Elementary Statistics Special Distributions

SQQS1013 Elementary Statistics
SPECIALSPECIAL
DISTRIBUTIONSDISTRIBUTIONS
5.1 DISCRETE DISTRIBUTION
• Consist of the values a random variable can assume and the corresponding
probabilities of the values.
• The probabilities are determined theoretically or by observation.
5.1.1 Binomial Probability Distribution
• Binomial Probability Distribution – the outcomes of a binomial experiment
and the corresponding probabilities of these outcomes.
• It is applied to find the probability that an outcome will occur x times in n
performances of experiment.
• For example:
 The probability of a defective laptop manufactured at a firm is 0.05 in a
random sample of ten.
 The probability of 8 packages will not arrive at its destination.
• To apply the binomial probability distribution, the random variable x must be
a discrete dichotomous random variable.
• Each repetition of the experiment must result in one of two possible
outcomes.
Characteristics of a Binomial Experiment
1. Each trial can have only two outcomes or outcomes that can be reduced to
two outcomes. These outcomes can be considered as either success or
failure.
2. There must be a fixed number of trials.
3. The outcomes of each trial must be independent of each other. In other
words, the outcome of one trial does not affect the outcome of another trial.
Chapter 5: Special Distributions 1

4. The probability of success is denoted by p and that of failure by q, and p +
q=1. The probabilities p and q remain constant for each trial.
5.1.1.1 Calculating Binomial Probability
There are two methods to calculate the binomial probability:
a) Using Binomial Probability Distribution Formula
• For a binomial experiment, the probability of exactly x successes in n trials
is given by the binomial formula:
P(x) = n
Cx px
qn-x
Where;
n = the total number of trials
p = probability of success
q = 1-p = probability of failure
x = number of successes in n trials
n-x = number of failures in n trials
Compute the probabilities of X successes, using the binomial formula; n= 6, X= 3,
p=0.03
Solution:
a) P(x=3) = 6
C3 (0.03)3
(1-0.03)6-3
= 6
C3 (0.03)3
(0.97)3
= 0.0005
A survey found that one out five Malaysian says he or she has visited a doctor in
any given month. If 10 people are selected at random, find the probability that
exactly 3 will have visited a doctor last month.
Solution:
Example 1
Example 2
FORMULA
ξ∆Σ λϖ
β
• The success does not mean that the
corresponding outcome is considered favorable
or desirable and vice versa
• The outcome to which the question refers is
called a success; the outcome to which it does
not refer is called a failure.

b) Using the Table of Binomial Probabilities
• The probabilities for a binomial experiment can also be read from the table of
binomial probabilities.
• For any number of trials n:
 The binomial probability distribution is symmetric if p = 0.5
 The binomial probability distribution is skewed to the right if p is less than
0.5
 The binomial probability distribution is skewed to the left if p is greater than
0.5
Determining P (x ≤ 3) for n = 6 and p = 0.30
The table of cumulative binomial probabilities (less than)
p
n x 0.10 0.15 0.20 0.25 0.30 0.35 0.40
6 0 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467
1 0.8857 0.7765 0.6554 0.5339 0.4202 0.3191 0.2333
2 0.9842 0.9527 0.9011 0.8306 0.7443 0.6471 0.5443
3 0.9987 0.9941 0.9830 0.9624 0.9295 0.8826 0.8208
4 0.9999 0.9996 0.9984 0.9954 0.9891 0.9777 0.9590
5 1.0000 1.0000 0.9999 0.9998 0.9993 0.9982 0.9959
6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
If you are using cumulative binomial probabilities table; P(X ≤ x); it is easier to calculate
various from of binomial distribution such as:
Equally P(X = x) = P(X ≤ x) - P(X ≤ x - 1) or use formula
At most P(X ≤ x) = P(X ≤ x) (directly from the table)
Less than P(X < x) = P(X ≤ x - 1)
At least P(X ≥ x) = 1 – P(X ≤ x - 1)
Greater than P(X > x) = 1 – P(X ≤ x)
From x1 to x2 P(x1 ≤ X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 - 1)
0
( ) (1 )
x
n r n r
r
r
P X x C p p −
=
≤ = −∑ p = 0.30
P (x ≤ 3) = 0.9295
n = 6
x ≤ 3
How to read the probability value from the table of binomial probabilities

Between x1 and x2 P(x1<X<x2) = P(X ≤ x2 - 1) - P(X ≤ x1)
Between x1 to x2 P(x1< X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 )
Compute the probability of X successes using the Binomial Table where n=2, p=0.30
and X ≤1
Solution:
P(X ≤1) = 0.9100
25% of all VCR manufactured by a large electronics company are defective. A quality
control inspector randomly selects three VCRs from the production line. What is the
probability that,
a) exactly one of these three VCRs is defective.
b) at least two of these VCRs are defective.
Solution:
Example 3
Example 4

EXERCISE 1
1. A burglar alarm system has 6 fail-safe components. The probability of each failing is
0.05. Find these probabilities:
a. Exactly 3 will fail
b. Less than 2 will fail
c. None will fail
2. A survey from Teenage Research Unlimited found that 30% of teenage consumers
receive their spending money from part-time jobs. If 5 teenagers are selected at
random, find the probability that at least 3 of them will have part-time jobs.
3. R. H Bruskin Associates Market Research found that 40% of Americans do not think
that having a college education is important to succeed in the business world. If a
random sample of five American is selected, find these probabilities.
a. Exactly two people will agree with that statement.
b. At most three people will agree with that statement
c. At least two people will agree with that statement
d. Fewer than three people will agree with that statement.
4. It was found that 60% of American victims of health care fraud are senior citizens. If
10 victims are randomly selected, find the probability that exactly 3 are senior citizens.
5. If 65% of the people in a community use the gym facilities in one year, find these
probabilities for a sample of 10 people
a) Exactly four people used the gym facilities.
b) At least six people not used the gym facilities
6. In a poll of 12 to 18 year old females conducted by Harris Interactive for the Gillette
Company, 40% of the young female said that they expected the US to have a female
president within 10 years. Supposed a random sample of 15 females from this age
group selected. Find the probabilities that of young female in this sample who expect
a female president within 10 years is;
a. at most 5
b. less than 4
c. at least 9
d. 6 to 9
e. in between 4 and 8

5.1.1.2 Mean and Standard Deviation of the Binomial Probability Distribution
The mean, variance and standard deviation of a binomial distribution are:
Mean = µ = np
Variance = σ2
= npq
Standard deviation = σ = npq
Where;
n = the total number of trials
p = probability of success
q = 1-p = probability of failure
Find the mean, variance and standard deviation for each of the values of n and p when
the conditions for the binomial distribution are met.
a. n=100, p=0.75
Solution:
a. µ = np = 100 (0.75) = 75
σ2
= npq = 100 (0.75) (0.25) = 18.75
σ = √npq = √18.75 = 4.33
The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of
all American births result in twins. If a random sample of 8000 births is taken, find the
mean, variance and standard deviation of the number of births that would result in
twins.
FORMULA
ξ∆Σ λϖ
β
Example 5
Example 6

EXERCISE 2
1. It has been reported that 83% of federal government employees use e-mail. If
sample of 200 federal government employees is selected, find the mean,
variance and standard deviation of the number who use e-mail.
2. A survey found that 25% of Malaysian watch movie at the cinema. Find the
mean, variance and standard deviation of the number of individuals who watch
movie at the cinema if a random sample of 1000 Malaysian is selected at the
Bintang Walk.
5.1.2 Poisson Distribution
• Derived from the French mathematician Simeon D. Poisson.
• A discrete probability distribution that is useful when n is large and p is small
and when the independent variables occur over a period of time or given
area or volume.
• Conditional to apply the Poisson Probability Distribution
 x is a discrete random variable
 The occurrences are random
 The occurrences are independent
• The following examples show the application of the Poisson probability
distribution.
 The number of accidents that occur on a highway given during a one-week
period.
 The number of customers entering a grocery store during a one-hour interval.
 The number of television sets sold at a department store during a given week.
 The number of typing errors per page.
 A certain type of fabric made contains an average 0.5 defects per 500 yards.

5.1.2.1 Calculating Poisson Probability
a) Using Poisson Probability Distribution Formula
According to the Poisson probability distribution, the probability of x occurrences in
an interval is:
P(X)=
!
x
e
x
λ
λ−
Where;
λ : mean number of occurrences in that interval
e : approximately 2.7183
Find probability P(X; λ), using the Poisson formula for P(5;4)
Solution:
P(X=5) =
4 5
(4 )
5!
e−
= 0.1563
If there are 200 typographical errors randomly distributed in a 500-page manuscript, find
the probability that a given page contains exactly three errors.
Solution:
FORMULA
ξ∆Σ λϖ
β
Example 7
Example 8

b) Using the Table of Poisson Probabilities
• The probabilities for a Poisson distribution can also be read from the table
of Poisson probabilities.
Determining P (x ≤ 3) for λ = 1.5
The table of cumulative Poisson probabilities (less than)
λ
x 1.1 1.2 1.3 1.4 1.5 1.6 1.7
0 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827
1 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932
2 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572
3 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068
4 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704
5 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920
6 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981
7 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996
If you are using cumulative Poisson probabilities table; P(X ≤ x); it is easier to
calculate various from of binomial distribution such as:
Chapter 5: Special Distributions
Equally P(X = x) = P(X ≤ x) - P(X ≤ x - 1) or using Poisson formula
At most P(X ≤ x) = P(X ≤ x) (directly from the table)
Less than P(X < x) = P(X ≤ x - 1)
At least P(X ≥ x) = 1 – P(X ≤ x - 1)
Greater than P(X > x) = 1 – P(X ≤ x)
From x1 to x2 P(x1 ≤ X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 - 1)
Between x1 and x2 P(x1<X<x2) = P(X ≤ x2 - 1) - P(X ≤ x1)
Between x1 to x2 P(x1< X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 )
9
How to read the probability value from the table of Poisson probabilities
λ = 1.50
( )
!
kx
k
e
P X x
k
λ
λ−
=
≤ =∑
x ≤ 3 P (x ≤ 3) = 0.9344

Find the probability P(X; λ); using Poisson table; P(10;7)
Solution:
P(X = 10) = P(X ≤ 10) – P(X ≤ 9)
= 0.9015 – 0.8305
= 0.0710
A sales firm receives on average three calls per hour on its toll-free number. For any
given hour, find the probability that it will receive the following:
a) at most 3 calls
b) at least 3 calls
c) five or more calls
d) between 1 to 4 calls in 2 hours
Solution:
λ = 3 calls per hour
a) P(X ≤ 3) = 0.6472
b) P(X ≥ 3) = 1 – P(X ≤ 2)
= 1 – 0.4232
= 0.5768
c) P(X ≥ 5) = 1 – P(X ≤ 4)
= 1 – 0.8153
= 0.1847
λ = 6 calls in 2 hours
d) P(1< X ≤ 4) = P(X ≤ 4) – P(X ≤ 1)
= 0.2851 – 0.0174
= 0.2677
Example 9
Example 10

EXERCISE 3
1. On average a household receives 2 telemarketing phone calls per week. Using the
Poisson distribution formula, find the probability that a randomly selected household
receives:
a. exactly six telemarketing phone calls during a given week.
b. less than three telemarketing phone calls in one month.
2. A washing machine in a LaundryMat breaks down an average of three times per
month. Using the Poisson probability distribution formula, find the probability that
this machine will have
a) exactly two break downs per month
b) at most one break down per month
c) no break downs in 2 months
3. In an airport between 1900 and 2000 hours, the number of airplane that land follows
a Poisson distribution with mean 0.9 per five minutes interval. Find the probability
that the number of plane that land is:
a) one or less between 1900 and 1905 hours
b) more than three between 1915 and 1930 hours
4. An average of 4.8 customers comes to Malaysia Savings and Loan every hour.
Find the probability that during a given hour, the number of customer will come is
a) exactly two
b) at most 2
c) none
d) more than 5 in half an hour
5. Sports Score Jay receives, on average, eight calls per hour requesting the latest
sports score. The distribution is Poisson in nature. For any randomly selected hour,
find the probability that the company will receive
a) at least eight calls
b) three or more calls
c) at most five calls.

5.1.2.2 Mean and Standard Deviation of the Poisson Probability
Distribution
Mean, µ = λ
Variance, σ2
= λ
Standard Deviation, σ = λ
An auto salesperson sells an average of 0.9 cars per day. Find the mean, variance
and standard deviation of cars sold per day by this salesperson.
Solution:
µ = λ = 0.9 σ2
= λ = 0.9
σ = √λ = √0.9 = 0.9487
An insurance salesperson sells an average of 1.4 policies per day.
a) Find the probability that this salesperson will sell no insurance policy on a
certain day.
b) Find the mean, variance and standard deviation of the probability this
salesperson will sell the policies per day.
Solution:
FORMULA
ξ∆Σ λϖ
β
Example 11
Example 12

5.2 CONTINUOUS DISTRIBUTION
• In Chapter 4, we defined a continuous random variable as a random variable
whose values are not countable.
• A continuous random variable can assume any value over an interval or
intervals because the number of values contained in any interval is infinite.
• The possible number of values that a continuous random variable can
assume is also infinite.
• Example of continuous random variables:
 The life of battery, heights of people, time taken to complete an examination,
amount of milk in a gallon, weights of babies, prices of houses.
• The probability distribution of continuous random variable has two
characteristics:
1. The probability that x assumes a value in any interval lies in the range 0 to 1.
Figure 1: Area under a curve between two points.
2. The total probability of all the (mutually exclusive) intervals within which x can
assume a value is 1.0.
Figure 2: Total area under a probability distribution.

5.2.1 Normal Distribution
• The normal distribution is the most important and most commonly used
among all of probability distributions.
• A large number of phenomena in the real world are normally distributed
either exactly or approximately.
• The normal probability distribution or the normal curve is a bell-shaped
(symmetric) curve.
• Its mean is denoted by µ while its standard deviation is denoted by σ.
Figure 3: Normal distribution with mean; µ and standard deviation; σ.
• A plotted normal probability distribution will gives a bell-shaped curve which
can be illustrated likes:
a) The total area under the curve is 1.0. b) The curve is symmetric about the
mean.
c)

c) The two tails of the curve extend indefinitely which means that the curve never
touch x- axis.
5.2.1.1 The Standard Normal Distribution
• Is a normal distribution with µ = 0 and σ = 1.
• The value under the curve indicates the proportion of area in each section.
(example figure 2; pg 13)
• The units for the standard normal distribution curve are denoted by z and
called the z values or z scores.
• The z value or z score is actually the number of standard deviation that a
particular x value is away from the mean.
• The area under a standard normal distribution curve is used to solve
practical application problems such as:
 finding the percentage of adult woman whose height is between 5 feet 4
inches and 5 feet 7 inches.

A. Finding areas under the standard normal distribution curve
• The standard normal distribution table lists the areas under the
standard normal curve to the left of z-values from –3.49 to 3.49.
• Although the z-values on the left side of the mean are negative, the area
under the curve is always positive.
Determine the area under the standard normal curve to the left of z = 1.95
How to read the probability value from the standard normal distribution table

Note:
z = 1.95 can be interpreted as area to the left of 1.95.
P(z < 1.95) = 0.9744 or P(z < 1.95) = P(z ≤ 1.95) = 0.9744
The probability that a continuous random variable assumes a single value is
zero. Therefore, P(z = 1.95) = 0.
Find the area under the standard normal curve:
a) To the left of z = 1.56
b) To the right of z = -1.32
c) From z = 0.85 to z = 1.95
Solution:
a) To the left of z = 1.56
P(z < 1.56) = 0.9406
b) To the right of z = -1.32
P(z > -1.32) = 1 – P(z < -1.32)
= 1 - 0.0934
= 0.9066
c) From z = 0.85 to z = 1.95
P(0.85 ≤ z ≤ 1.95)
= P(z ≤ 1.95) – P(z ≤ 0.85)
Example 13
0 1.56
-1.32 0
0.85 1.95

= 0.9744 – 0.8023
= 0.1721
EXERCISE 4
Find the area under the standard normal curve:
a) To the left of z = -2.87
b) To the right of z = 2.45
c) Between z = -2.15 and z=1.67
B. Converting an x Value to a z Value
For a normal variable x, a particular value of x can be converted to its
corresponding z value by using the formula:
z =
x µ
σ
−
Where µ and σ are the mean and standard deviation of the normal distribution of
x, respectively.
FORMULA
ξ∆Σ λϖ
β
Remember!
The z value for the mean of a normal
distribution is always zero.

Let x be a continuous random variable that has a normal distribution with a mean of 50
and a standard deviation of 10. Convert the following x values to z values.
a) 55 b) 35
Solution:
For the given normal distribution, µ = 50 and σ =10
a) The z value for x = 55 is computes as follows:
b) z = 35 – 50 =
10
Let x be a continuous random variable that is normally distributed with a mean of 65 and
a standard deviation of 15. Find the probability that x can assumes a value:
a) less than 43
b) greater than 74
c) between 56 and 71
Solution:
Example 14
Example 15

EXERCISE 5
1. Let x denote the time takes to run a road race. Supposed x is approximately
normally distributed with mean of 190 minutes and standard deviation of 21 minutes.
If one runner is selected at random, what is the probability that this runner will
complete this road race?
a) in less than 150 minutes
b) in 205 to 245 minutes
2. The mean number of hours a student spends on the computer is 3.1 hours per day.
Assume the standard deviation is 0.5 hour. Find the percentage of students who
spend less than 3.5 hours on the computer. Assume the variable is normally
distributed.
3. The score of 6000 candidates in a certain examination are found to be approximately
normal distributed with a mean of 55 and a standard deviation of 10:
a) If a score of 75 or more is required for passing the distinction, estimate the
number of grades with distinction.
b) Calculate the probability that a candidate selected at random has a score
between 45 and 65.
5.3 INTRODUCTION TO t- DISTRIBUTION
• The t distribution is very similar to the standardized normal distribution.
• Both distributions are bell-shaped and symmetrical.
• However, the t distribution has more area in the tails and less in the center
than does the standardized normal distribution.
• This is because σ is unknown and S is used to estimate it.
• Because the value of σ is uncertain, the values of t that are observed will be
more variable than for Z.
Standard normal
t distribution for 5
degrees of freedom

• As the number of degrees of freedom increases, the t distribution gradually
approaches the standardized normal distribution until the two are virtually
identical.
• This happens because S becomes a better estimate of σ as the sample size
gets larger.
• With a sample size of about 120 or more, S estimates σ precisely enough
that there is little difference between the t and Z distributions.
• For this reason, most statisticians use Z instead of t when the sample size is
greater than 120.
EXERCISE 6
1. 10 % of the bulbs produced by a factory are defective. A sample of 5 bulbs is
selected randomly and tested for defect. Find the probability that
a) two bulbs are defective
b) at least one bulb is defective
2. In a university, 20 percent of the students fail the statistic test. If 20 students from the
university are interviewed, what is the probability of getting:
a) less than 3 students who fail the test
b) exactly 4 students who fail the test
3. A financial institution in Kuala Lumpur has offer a job as a risk analyst. For the
minimum qualification, the applicant must seats for writing test. Based on the
management experience, 40% of the applicants will pass the test and qualified for
the interview session. There are 20 applicants who have applied for the jobs. Find
the probability that there are more than 50% applicants will pass the test.

4. An Elementary Statistic class has 75 members. If there is a 12% absentee rate per
class meeting, find the mean, variance and standard deviation of the number of
students who will be absent from each class.
5. Before an umbrella leaves the factory, it is given a quality control check. The
probability that an umbrella contains zero, one or two defects is 0.88, 0.08 and 0.04
respectively. In a sample of 16 umbrellas, find the probability that:
a) 9 will have no defect
b) 4 will have one defect
c) 3 will have two defect
6. En. Rostam is a credit officer at the Trust Bank. Based on his experience, he
estimates that an average he will receive the loan application in a week is 3
applications. Find the probability that:
a) he receives none loan application in a week
b) he receives 2 until 5 loan applications in a week
c) at least 5 loan applications he receives in 14 days.
7. A bookstore owner examines 5 books from each lot of 25 to check for missing pages.
If he finds at least two books with missing pages, the entire lot is returned. If indeed,
there are five books with missing pages, find the probability that the lot will be
returned.
8. The numbers of customers who enter shop ABC independent of one another and at
random intervals follow a Poisson distribution with an average rate 42 customers per
hour. Find the probability that:
a) no customer enter the shop during a particular 1 minutes interval
b) at least 4 customers enter the shop during a particular 5 minutes interval
c) between 2 and 6 customers enter the shop during a particular 10-minute
interval.
9. One research has been conducted by Student’s Affair Department of Menara
University about the CGPA of final semester student for 2010/2011 session. The
outcome of the research showed that the CGPA of the student is normally distributed
with mean 2.80 and standard deviation of 0.40. If one final semester student has
been selected at random;
a) Calculate the probability that the student’s CGPA is between 2.00 and 3.00
b) Find the percentage that the student’s CGPA is less than 2.00
c) Calculate the probability that the student’s CGPA is at least 3.00

d) Calculate the probability that the student’s CGPA is more than 3.70 (first
class honors). If the university has 1000 final semester students, find the
number of first class honors student.
10. Encik Ahmad works as a lawyer at his own law firm which situated at Bandar
Kenangan. He drives his car to go to his workplace and back everyday. The
estimated time taken for the journey is normal distributed with mean 24 minutes
and standard deviation is 3.8 minutes.
a) Calculate the probability of estimated time taken by Encik Ahmad to go to
his workplace and back is at least half an hour.
b) Find the probability of estimated time taken by Encik Ahmad to go to his
workplace and back is 20 minutes to 25 minutes
c) Find the percentage that the estimated time taken by Encik Ahmad to go
to his workplace and back is more than 25 minutes.
d) Find the probability of estimated time taken by Encik Ahmad to go to his
workplace and back is less than 10 minutes. Explain.
11. Assume X is timing for a runner to finish his 2 km run. Given X is normally
distributed with mean 15 minutes and standard deviation is 3 minutes. If one
runner is selected at random, find the probability that the runner can finish his 2km
run in time;
a) Less than 13 minutes
b) Not more than 16 minutes
c) Within 14 minutes and 17 minutes.
12. Given the systolic blood pressure for the obesity group has mean 132 mmHg and
standard deviation 8 mmHg. Assume the variable normally distributed, find the
probability an obese person that has been selected at random have a systolic
blood pressure:
a) More than 130 mmHg.
b) Less than 140 mmHg.
c) Between 131 mmHg and 136 mmHg.
13. The number of passenger for domestic flight from Alor Setar to Kuala Lumpur is
normally distributed with mean 80 and standard deviation 12. If one domestic
flight is selected at random, find the probability the plain has a passenger:

a) less than 90 passengers
b) at least 75 passengers
c) between 79 to 95 passengers

BINOMIAL DISTRIBUTION
p
n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
1 0 0.9000 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000
1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 0 0.8100 0.7225 0.6400 0.5625 0.4900 0.4225 0.3600 0.3025 0.2500
1 0.9900 0.9775 0.9600 0.9375 0.9100 0.8775 0.8400 0.7975 0.7500
2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3 0 0.7290 0.6141 0.5120 0.4219 0.3430 0.2746 0.2160 0.1664 0.1250
1 0.9720 0.9393 0.8960 0.8438 0.7840 0.7183 0.6480 0.5748 0.5000
2 0.9990 0.9966 0.9920 0.9844 0.9730 0.9571 0.9360 0.9089 0.8750
3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4 0 0.6561 0.5220 0.4096 0.3164 0.2401 0.1785 0.1296 0.0915 0.0625
1 0.9477 0.8905 0.8192 0.7383 0.6517 0.5630 0.4752 0.3910 0.3125
2 0.9963 0.9880 0.9728 0.9492 0.9163 0.8735 0.8208 0.7585 0.6875
3 0.9999 0.9995 0.9984 0.9961 0.9919 0.9850 0.9744 0.9590 0.9375
4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
5 0 0.5905 0.4437 0.3277 0.2373 0.1681 0.1160 0.0778 0.0503 0.0313
1 0.9185 0.8352 0.7373 0.6328 0.5282 0.4284 0.3370 0.2562 0.1875
2 0.9914 0.9734 0.9421 0.8965 0.8369 0.7648 0.6826 0.5931 0.5000
3 0.9995 0.9978 0.9933 0.9844 0.9692 0.9460 0.9130 0.8688 0.8125
4 1.0000 0.9999 0.9997 0.9990 0.9976 0.9947 0.9898 0.9815 0.9688
5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
6 0 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0277 0.0156
1 0.8857 0.7765 0.6554 0.5339 0.4202 0.3191 0.2333 0.1636 0.1094
2 0.9842 0.9527 0.9011 0.8306 0.7443 0.6471 0.5443 0.4415 0.3438
3 0.9987 0.9941 0.9830 0.9624 0.9295 0.8826 0.8208 0.7447 0.6563
4 0.9999 0.9996 0.9984 0.9954 0.9891 0.9777 0.9590 0.9308 0.8906
5 1.0000 1.0000 0.9999 0.9998 0.9993 0.9982 0.9959 0.9917 0.9844
6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
7 0 0.4783 0.3206 0.2097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078
1 0.8503 0.7166 0.5767 0.4449 0.3294 0.2338 0.1586 0.1024 0.0625
2 0.9743 0.9262 0.8520 0.7564 0.6471 0.5323 0.4199 0.3164 0.2266
3 0.9973 0.9879 0.9667 0.9294 0.8740 0.8002 0.7102 0.6083 0.5000
4 0.9998 0.9988 0.9953 0.9871 0.9712 0.9444 0.9037 0.8471 0.7734
5 1.0000 0.9999 0.9996 0.9987 0.9962 0.9910 0.9812 0.9643 0.9375
6 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9984 0.9963 0.9922
7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
continue;
Chapter 5: Special Distribution
Binomial Distribution Table
25
0
( ) (1 )
x
n r n r
r
r
P X x C p p −
=
≤ = −∑

p
26

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
8 0 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.0039
1 0.8131 0.6572 0.5033 0.3671 0.2553 0.1691 0.1064 0.0632 0.0352
2 0.9619 0.8948 0.7969 0.6785 0.5518 0.4278 0.3154 0.2201 0.1445
3 0.9950 0.9786 0.9437 0.8862 0.8059 0.7064 0.5941 0.4770 0.3633
4 0.9996 0.9971 0.9896 0.9727 0.9420 0.8939 0.8263 0.7396 0.6367
5 1.0000 0.9998 0.9988 0.9958 0.9887 0.9747 0.9502 0.9115 0.8555
6 1.0000 1.0000 0.9999 0.9996 0.9987 0.9964 0.9915 0.9819 0.9648
7 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9993 0.9983 0.9961
8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
9 0 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.0020
1 0.7748 0.5995 0.4362 0.3003 0.1960 0.1211 0.0705 0.0385 0.0195
2 0.9470 0.8591 0.7382 0.6007 0.4628 0.3373 0.2318 0.1495 0.0898
3 0.9917 0.9661 0.9144 0.8343 0.7297 0.6089 0.4826 0.3614 0.2539
4 0.9991 0.9944 0.9804 0.9511 0.9012 0.8283 0.7334 0.6214 0.5000
5 0.9999 0.9994 0.9969 0.9900 0.9747 0.9464 0.9006 0.8342 0.7461
6 1.0000 1.0000 0.9997 0.9987 0.9957 0.9888 0.9750 0.9502 0.9102
7 1.0000 1.0000 1.0000 0.9999 0.9996 0.9986 0.9962 0.9909 0.9805
8 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9992 0.9980
9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
10 0 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010
1 0.7361 0.5443 0.3758 0.2440 0.1493 0.0860 0.0464 0.0233 0.0107
2 0.9298 0.8202 0.6778 0.5256 0.3828 0.2616 0.1673 0.0996 0.0547
3 0.9872 0.9500 0.8791 0.7759 0.6496 0.5138 0.3823 0.2660 0.1719
4 0.9984 0.9901 0.9672 0.9219 0.8497 0.7515 0.6331 0.5044 0.3770
5 0.9999 0.9986 0.9936 0.9803 0.9527 0.9051 0.8338 0.7384 0.6230
6 1.0000 0.9999 0.9991 0.9965 0.9894 0.9740 0.9452 0.8980 0.8281
7 1.0000 1.0000 0.9999 0.9996 0.9984 0.9952 0.9877 0.9726 0.9453
8 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9983 0.9955 0.9893
9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
11 0 0.3138 0.1673 0.0859 0.0422 0.0198 0.0088 0.0036 0.0014 0.0005
1 0.6974 0.4922 0.3221 0.1971 0.1130 0.0606 0.0302 0.0139 0.0059
2 0.9104 0.7788 0.6174 0.4552 0.3127 0.2001 0.1189 0.0652 0.0327
3 0.9815 0.9306 0.8389 0.7133 0.5696 0.4256 0.2963 0.1911 0.1133
4 0.9972 0.9841 0.9496 0.8854 0.7897 0.6683 0.5328 0.3971 0.2744
5 0.9997 0.9973 0.9883 0.9657 0.9218 0.8513 0.7535 0.6331 0.5000
6 1.0000 0.9997 0.9980 0.9924 0.9784 0.9499 0.9006 0.8262 0.7256
7 1.0000 1.0000 0.9998 0.9988 0.9957 0.9878 0.9707 0.9390 0.8867
8 1.0000 1.0000 1.0000 0.9999 0.9994 0.9980 0.9941 0.9852 0.9673
9 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9993 0.9978 0.9941
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9995
11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Continue;
p
27

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
12 0 0.2824 0.1422 0.0687 0.0317 0.0138 0.0057 0.0022 0.0008 0.0002
1 0.6590 0.4435 0.2749 0.1584 0.0850 0.0424 0.0196 0.0083 0.0032
2 0.8891 0.7358 0.5583 0.3907 0.2528 0.1513 0.0834 0.0421 0.0193
3 0.9744 0.9078 0.7946 0.6488 0.4925 0.3467 0.2253 0.1345 0.0730
4 0.9957 0.9761 0.9274 0.8424 0.7237 0.5833 0.4382 0.3044 0.1938
5 0.9995 0.9954 0.9806 0.9456 0.8822 0.7873 0.6652 0.5269 0.3872
6 0.9999 0.9993 0.9961 0.9857 0.9614 0.9154 0.8418 0.7393 0.6128
7 1.0000 0.9999 0.9994 0.9972 0.9905 0.9745 0.9427 0.8883 0.8062
8 1.0000 1.0000 0.9999 0.9996 0.9983 0.9944 0.9847 0.9644 0.9270
9 1.0000 1.0000 1.0000 1.0000 0.9998 0.9992 0.9972 0.9921 0.9807
10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9989 0.9968
11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998
12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
13 0 0.2542 0.1209 0.0550 0.0238 0.0097 0.0037 0.0013 0.0004 0.0001
1 0.6213 0.3983 0.2336 0.1267 0.0637 0.0296 0.0126 0.0049 0.0017
2 0.8661 0.6920 0.5017 0.3326 0.2025 0.1132 0.0579 0.0269 0.0112
3 0.9658 0.8820 0.7473 0.5843 0.4206 0.2783 0.1686 0.0929 0.0461
4 0.9935 0.9658 0.9009 0.7940 0.6543 0.5005 0.3530 0.2279 0.1334
5 0.9991 0.9925 0.9700 0.9198 0.8346 0.7159 0.5744 0.4268 0.2905
6 0.9999 0.9987 0.9930 0.9757 0.9376 0.8705 0.7712 0.6437 0.5000
7 1.0000 0.9998 0.9988 0.9944 0.9818 0.9538 0.9023 0.8212 0.7095
8 1.0000 1.0000 0.9998 0.9990 0.9960 0.9874 0.9679 0.9302 0.8666
9 1.0000 1.0000 1.0000 0.9999 0.9993 0.9975 0.9922 0.9797 0.9539
10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9987 0.9959 0.9888
11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9983
12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
14 0 0.2288 0.1028 0.0440 0.0178 0.0068 0.0024 0.0008 0.0002 0.0001
1 0.5846 0.3567 0.1979 0.1010 0.0475 0.0205 0.0081 0.0029 0.0009
2 0.8416 0.6479 0.4481 0.2811 0.1608 0.0839 0.0398 0.0170 0.0065
3 0.9559 0.8535 0.6982 0.5213 0.3552 0.2205 0.1243 0.0632 0.0287
4 0.9908 0.9533 0.8702 0.7415 0.5842 0.4227 0.2793 0.1672 0.0898
5 0.9985 0.9885 0.9561 0.8883 0.7805 0.6405 0.4859 0.3373 0.2120
6 0.9998 0.9978 0.9884 0.9617 0.9067 0.8164 0.6925 0.5461 0.3953
7 1.0000 0.9997 0.9976 0.9897 0.9685 0.9247 0.8499 0.7414 0.6047
8 1.0000 1.0000 0.9996 0.9978 0.9917 0.9757 0.9417 0.8811 0.7880
9 1.0000 1.0000 1.0000 0.9997 0.9983 0.9940 0.9825 0.9574 0.9102
10 1.0000 1.0000 1.0000 1.0000 0.9998 0.9989 0.9961 0.9886 0.9713
11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9978 0.9935
12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991
13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Continue;
p
28

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
15 0 0.2059 0.0874 0.0352 0.0134 0.0047 0.0016 0.0005 0.0001 0.0000
1 0.5490 0.3186 0.1671 0.0802 0.0353 0.0142 0.0052 0.0017 0.0005
2 0.8159 0.6042 0.3980 0.2361 0.1268 0.0617 0.0271 0.0107 0.0037
3 0.9444 0.8227 0.6482 0.4613 0.2969 0.1727 0.0905 0.0424 0.0176
4 0.9873 0.9383 0.8358 0.6865 0.5155 0.3519 0.2173 0.1204 0.0592
5 0.9978 0.9832 0.9389 0.8516 0.7216 0.5643 0.4032 0.2608 0.1509
6 0.9997 0.9964 0.9819 0.9434 0.8689 0.7548 0.6098 0.4522 0.3036
7 1.0000 0.9994 0.9958 0.9827 0.9500 0.8868 0.7869 0.6535 0.5000
8 1.0000 0.9999 0.9992 0.9958 0.9848 0.9578 0.9050 0.8182 0.6964
9 1.0000 1.0000 0.9999 0.9992 0.9963 0.9876 0.9662 0.9231 0.8491
10 1.0000 1.0000 1.0000 0.9999 0.9993 0.9972 0.9907 0.9745 0.9408
11 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9981 0.9937 0.9824
12 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9989 0.9963
13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
16 0 0.1853 0.0743 0.0281 0.0100 0.0033 0.0010 0.0003 0.0001 0.0000
1 0.5147 0.2839 0.1407 0.0635 0.0261 0.0098 0.0033 0.0010 0.0003
2 0.7892 0.5614 0.3518 0.1971 0.0994 0.0451 0.0183 0.0066 0.0021
3 0.9316 0.7899 0.5981 0.4050 0.2459 0.1339 0.0651 0.0281 0.0106
4 0.9830 0.9209 0.7982 0.6302 0.4499 0.2892 0.1666 0.0853 0.0384
5 0.9967 0.9765 0.9183 0.8103 0.6598 0.4900 0.3288 0.1976 0.1051
6 0.9995 0.9944 0.9733 0.9204 0.8247 0.6881 0.5272 0.3660 0.2272
7 0.9999 0.9989 0.9930 0.9729 0.9256 0.8406 0.7161 0.5629 0.4018
8 1.0000 0.9998 0.9985 0.9925 0.9743 0.9329 0.8577 0.7441 0.5982
9 1.0000 1.0000 0.9998 0.9984 0.9929 0.9771 0.9417 0.8759 0.7728
10 1.0000 1.0000 1.0000 0.9997 0.9984 0.9938 0.9809 0.9514 0.8949
11 1.0000 1.0000 1.0000 1.0000 0.9997 0.9987 0.9951 0.9851 0.9616
12 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9991 0.9965 0.9894
13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9979
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
17 0 0.1668 0.0631 0.0225 0.0075 0.0023 0.0007 0.0002 0.0000 0.0000
1 0.4818 0.2525 0.1182 0.0501 0.0193 0.0067 0.0021 0.0006 0.0001
2 0.7618 0.5198 0.3096 0.1637 0.0774 0.0327 0.0123 0.0041 0.0012
3 0.9174 0.7556 0.5489 0.3530 0.2019 0.1028 0.0464 0.0184 0.0064
4 0.9779 0.9013 0.7582 0.5739 0.3887 0.2348 0.1260 0.0596 0.0245
5 0.9953 0.9681 0.8943 0.7653 0.5968 0.4197 0.2639 0.1471 0.0717
6 0.9992 0.9917 0.9623 0.8929 0.7752 0.6188 0.4478 0.2902 0.1662
7 0.9999 0.9983 0.9891 0.9598 0.8954 0.7872 0.6405 0.4743 0.3145
8 1.0000 0.9997 0.9974 0.9876 0.9597 0.9006 0.8011 0.6626 0.5000
Continue;
p
29

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
9 1.0000 1.0000 0.9995 0.9969 0.9873 0.9617 0.9081 0.8166 0.6855
10 1.0000 1.0000 0.9999 0.9994 0.9968 0.9880 0.9652 0.9174 0.8338
11 1.0000 1.0000 1.0000 0.9999 0.9993 0.9970 0.9894 0.9699 0.9283
12 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9975 0.9914 0.9755
13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9981 0.9936
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9988
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
18 0 0.1501 0.0536 0.0180 0.0056 0.0016 0.0004 0.0001 0.0000 0.0000
1 0.4503 0.2241 0.0991 0.0395 0.0142 0.0046 0.0013 0.0003 0.0001
2 0.7338 0.4797 0.2713 0.1353 0.0600 0.0236 0.0082 0.0025 0.0007
3 0.9018 0.7202 0.5010 0.3057 0.1646 0.0783 0.0328 0.0120 0.0038
4 0.9718 0.8794 0.7164 0.5187 0.3327 0.1886 0.0942 0.0411 0.0154
5 0.9936 0.9581 0.8671 0.7175 0.5344 0.3550 0.2088 0.1077 0.0481
6 0.9988 0.9882 0.9487 0.8610 0.7217 0.5491 0.3743 0.2258 0.1189
7 0.9998 0.9973 0.9837 0.9431 0.8593 0.7283 0.5634 0.3915 0.2403
8 1.0000 0.9995 0.9957 0.9807 0.9404 0.8609 0.7368 0.5778 0.4073
9 1.0000 0.9999 0.9991 0.9946 0.9790 0.9403 0.8653 0.7473 0.5927
10 1.0000 1.0000 0.9998 0.9988 0.9939 0.9788 0.9424 0.8720 0.7597
11 1.0000 1.0000 1.0000 0.9998 0.9986 0.9938 0.9797 0.9463 0.8811
12 1.0000 1.0000 1.0000 1.0000 0.9997 0.9986 0.9942 0.9817 0.9519
13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9987 0.9951 0.9846
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9990 0.9962
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
19 0 0.1351 0.0456 0.0144 0.0042 0.0011 0.0003 0.0001 0.0000 0.0000
1 0.4203 0.1985 0.0829 0.0310 0.0104 0.0031 0.0008 0.0002 0.0000
2 0.7054 0.4413 0.2369 0.1113 0.0462 0.0170 0.0055 0.0015 0.0004
3 0.8850 0.6841 0.4551 0.2631 0.1332 0.0591 0.0230 0.0077 0.0022
4 0.9648 0.8556 0.6733 0.4654 0.2822 0.1500 0.0696 0.0280 0.0096
5 0.9914 0.9463 0.8369 0.6678 0.4739 0.2968 0.1629 0.0777 0.0318
6 0.9983 0.9837 0.9324 0.8251 0.6655 0.4812 0.3081 0.1727 0.0835
7 0.9997 0.9959 0.9767 0.9225 0.8180 0.6656 0.4878 0.3169 0.1796
8 1.0000 0.9992 0.9933 0.9713 0.9161 0.8145 0.6675 0.4940 0.3238
9 1.0000 0.9999 0.9984 0.9911 0.9674 0.9125 0.8139 0.6710 0.5000
10 1.0000 1.0000 0.9997 0.9977 0.9895 0.9653 0.9115 0.8159 0.6762
11 1.0000 1.0000 1.0000 0.9995 0.9972 0.9886 0.9648 0.9129 0.8204
12 1.0000 1.0000 1.0000 0.9999 0.9994 0.9969 0.9884 0.9658 0.9165
13 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9969 0.9891 0.9682
Continue:
p
30

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9972 0.9904
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9978
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
20 0 0.1216 0.0388 0.0115 0.0032 0.0008 0.0002 0.0000 0.0000 0.0000
1 0.3917 0.1756 0.0692 0.0243 0.0076 0.0021 0.0005 0.0001 0.0000
2 0.6769 0.4049 0.2061 0.0913 0.0355 0.0121 0.0036 0.0009 0.0002
3 0.8670 0.6477 0.4114 0.2252 0.1071 0.0444 0.0160 0.0049 0.0013
4 0.9568 0.8298 0.6296 0.4148 0.2375 0.1182 0.0510 0.0189 0.0059
5 0.9887 0.9327 0.8042 0.6172 0.4164 0.2454 0.1256 0.0553 0.0207
6 0.9976 0.9781 0.9133 0.7858 0.6080 0.4166 0.2500 0.1299 0.0577
7 0.9996 0.9941 0.9679 0.8982 0.7723 0.6010 0.4159 0.2520 0.1316
8 0.9999 0.9987 0.9900 0.9591 0.8867 0.7624 0.5956 0.4143 0.2517
9 1.0000 0.9998 0.9974 0.9861 0.9520 0.8782 0.7553 0.5914 0.4119
10 1.0000 1.0000 0.9994 0.9961 0.9829 0.9468 0.8725 0.7507 0.5881
11 1.0000 1.0000 0.9999 0.9991 0.9949 0.9804 0.9435 0.8692 0.7483
12 1.0000 1.0000 1.0000 0.9998 0.9987 0.9940 0.9790 0.9420 0.8684
13 1.0000 1.0000 1.0000 1.0000 0.9997 0.9985 0.9935 0.9786 0.9423
14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9936 0.9793
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9985 0.9941
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9987
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
31

POISSON DISTRIBUTION
λ
x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066 0.3679
1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.7725 0.7358
2 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.9371 0.9197
3 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.9865 0.9810
4 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.9977 0.9963
5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9994
6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496 0.1353
1 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932 0.4628 0.4337 0.4060
2 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572 0.7306 0.7037 0.6767
3 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068 0.8913 0.8747 0.8571
4 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704 0.9636 0.9559 0.9473
5 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920 0.9896 0.9868 0.9834
6 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981 0.9974 0.9966 0.9955
7 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.9992 0.9989
8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9998
9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
0 0.1225 0.1108 0.1003 0.0907 0.0821 0.0743 0.0672 0.0608 0.0550 0.0498
1 0.3796 0.3546 0.3309 0.3084 0.2873 0.2674 0.2487 0.2311 0.2146 0.1991
2 0.6496 0.6227 0.5960 0.5697 0.5438 0.5184 0.4936 0.4695 0.4460 0.4232
3 0.8386 0.8194 0.7993 0.7787 0.7576 0.7360 0.7141 0.6919 0.6696 0.6472
4 0.9379 0.9275 0.9162 0.9041 0.8912 0.8774 0.8629 0.8477 0.8318 0.8153
5 0.9796 0.9751 0.9700 0.9643 0.9580 0.9510 0.9433 0.9349 0.9258 0.9161
6 0.9941 0.9925 0.9906 0.9884 0.9858 0.9828 0.9794 0.9756 0.9713 0.9665
7 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.9919 0.9901 0.9881
8 0.9997 0.9995 0.9994 0.9991 0.9989 0.9985 0.9981 0.9976 0.9969 0.9962
9 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.9991 0.9989
10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997
11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999
12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
Poisson Distribution Table
32
0
( )
!
kx
k
e
P X x
k
λ
λ−
=
≤ =∑

x 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
0 0.0450 0.0408 0.0369 0.0334 0.0302 0.0273 0.0247 0.0224 0.0202 0.0183
1 0.1847 0.1712 0.1586 0.1468 0.1359 0.1257 0.1162 0.1074 0.0992 0.0916
2 0.4012 0.3799 0.3594 0.3397 0.3208 0.3027 0.2854 0.2689 0.2531 0.2381
3 0.6248 0.6025 0.5803 0.5584 0.5366 0.5152 0.4942 0.4735 0.4532 0.4335
4 0.7982 0.7806 0.7626 0.7442 0.7254 0.7064 0.6872 0.6678 0.6484 0.6288
5 0.9057 0.8946 0.8829 0.8705 0.8576 0.8441 0.8301 0.8156 0.8006 0.7851
6 0.9612 0.9554 0.9490 0.9421 0.9347 0.9267 0.9182 0.9091 0.8995 0.8893
7 0.9858 0.9832 0.9802 0.9769 0.9733 0.9692 0.9648 0.9599 0.9546 0.9489
8 0.9953 0.9943 0.9931 0.9917 0.9901 0.9883 0.9863 0.9840 0.9815 0.9786
9 0.9986 0.9982 0.9978 0.9973 0.9967 0.9960 0.9952 0.9942 0.9931 0.9919
10 0.9996 0.9995 0.9994 0.9992 0.9990 0.9987 0.9984 0.9981 0.9977 0.9972
11 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 0.9991
12 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997
13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0
0 0.0166 0.0150 0.0136 0.0123 0.0111 0.0101 0.0091 0.0082 0.0074 0.0067
1 0.0845 0.0780 0.0719 0.0663 0.0611 0.0563 0.0518 0.0477 0.0439 0.0404
2 0.2238 0.2102 0.1974 0.1851 0.1736 0.1626 0.1523 0.1425 0.1333 0.1247
3 0.4142 0.3954 0.3772 0.3594 0.3423 0.3257 0.3097 0.2942 0.2793 0.2650
4 0.6093 0.5898 0.5704 0.5512 0.5321 0.5132 0.4946 0.4763 0.4582 0.4405
5 0.7693 0.7531 0.7367 0.7199 0.7029 0.6858 0.6684 0.6510 0.6335 0.6160
6 0.8786 0.8675 0.8558 0.8436 0.8311 0.8180 0.8046 0.7908 0.7767 0.7622
7 0.9427 0.9361 0.9290 0.9214 0.9134 0.9049 0.8960 0.8867 0.8769 0.8666
8 0.9755 0.9721 0.9683 0.9642 0.9597 0.9549 0.9497 0.9442 0.9382 0.9319
9 0.9905 0.9889 0.9871 0.9851 0.9829 0.9805 0.9778 0.9749 0.9717 0.9682
10 0.9966 0.9959 0.9952 0.9943 0.9933 0.9922 0.9910 0.9896 0.9880 0.9863
11 0.9989 0.9986 0.9983 0.9980 0.9976 0.9971 0.9966 0.9960 0.9953 0.9945
12 0.9997 0.9996 0.9995 0.9993 0.9992 0.9990 0.9988 0.9986 0.9983 0.9980
13 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.9994 0.9993
14 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
0 0.0061 0.0055 0.0050 0.0045 0.0041 0.0037 0.0033 0.0030 0.0027 0.0025
1 0.0372 0.0342 0.0314 0.0289 0.0266 0.0244 0.0224 0.0206 0.0189 0.0174
2 0.1165 0.1088 0.1016 0.0948 0.0884 0.0824 0.0768 0.0715 0.0666 0.0620
3 0.2513 0.2381 0.2254 0.2133 0.2017 0.1906 0.1800 0.1700 0.1604 0.1512
4 0.4231 0.4061 0.3895 0.3733 0.3575 0.3422 0.3272 0.3127 0.2987 0.2851
5 0.5984 0.5809 0.5635 0.5461 0.5289 0.5119 0.4950 0.4783 0.4619 0.4457
6 0.7474 0.7324 0.7171 0.7017 0.6860 0.6703 0.6544 0.6384 0.6224 0.6063
λ
33

x 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
7 0.8560 0.8449 0.8335 0.8217 0.8095 0.7970 0.7841 0.7710 0.7576 0.7440
8 0.9252 0.9181 0.9106 0.9027 0.8944 0.8857 0.8766 0.8672 0.8574 0.8472
9 0.9644 0.9603 0.9559 0.9512 0.9462 0.9409 0.9352 0.9292 0.9228 0.9161
10 0.9844 0.9823 0.9800 0.9775 0.9747 0.9718 0.9686 0.9651 0.9614 0.9574
11 0.9937 0.9927 0.9916 0.9904 0.9890 0.9875 0.9859 0.9841 0.9821 0.9799
12 0.9976 0.9972 0.9967 0.9962 0.9955 0.9949 0.9941 0.9932 0.9922 0.9912
13 0.9992 0.9990 0.9988 0.9986 0.9983 0.9980 0.9977 0.9973 0.9969 0.9964
14 0.9997 0.9997 0.9996 0.9995 0.9994 0.9993 0.9991 0.9990 0.9988 0.9986
15 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9996 0.9996 0.9995
16 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0
0 0.0022 0.0020 0.0018 0.0017 0.0015 0.0014 0.0012 0.0011 0.0010 0.0009
1 0.0159 0.0146 0.0134 0.0123 0.0113 0.0103 0.0095 0.0087 0.0080 0.0073
2 0.0577 0.0536 0.0498 0.0463 0.0430 0.0400 0.0371 0.0344 0.0320 0.0296
3 0.1425 0.1342 0.1264 0.1189 0.1118 0.1052 0.0988 0.0928 0.0871 0.0818
4 0.2719 0.2592 0.2469 0.2351 0.2237 0.2127 0.2022 0.1920 0.1823 0.1730
5 0.4298 0.4141 0.3988 0.3837 0.3690 0.3547 0.3406 0.3270 0.3137 0.3007
6 0.5902 0.5742 0.5582 0.5423 0.5265 0.5108 0.4953 0.4799 0.4647 0.4497
7 0.7301 0.7160 0.7017 0.6873 0.6728 0.6581 0.6433 0.6285 0.6136 0.5987
8 0.8367 0.8259 0.8148 0.8033 0.7916 0.7796 0.7673 0.7548 0.7420 0.7291
9 0.9090 0.9016 0.8939 0.8858 0.8774 0.8686 0.8596 0.8502 0.8405 0.8305
10 0.9531 0.9486 0.9437 0.9386 0.9332 0.9274 0.9214 0.9151 0.9084 0.9015
11 0.9776 0.9750 0.9723 0.9693 0.9661 0.9627 0.9591 0.9552 0.9510 0.9467
12 0.9900 0.9887 0.9873 0.9857 0.9840 0.9821 0.9801 0.9779 0.9755 0.9730
13 0.9958 0.9952 0.9945 0.9937 0.9929 0.9920 0.9909 0.9898 0.9885 0.9872
14 0.9984 0.9981 0.9978 0.9974 0.9970 0.9966 0.9961 0.9956 0.9950 0.9943
15 0.9994 0.9993 0.9992 0.9990 0.9988 0.9986 0.9984 0.9982 0.9979 0.9976
16 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994 0.9993 0.9992 0.9990
17 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996
18 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0
0 0.0008 0.0007 0.0007 0.0006 0.0006 0.0005 0.0005 0.0004 0.0004 0.0003
1 0.0067 0.0061 0.0056 0.0051 0.0047 0.0043 0.0039 0.0036 0.0033 0.0030
2 0.0275 0.0255 0.0236 0.0219 0.0203 0.0188 0.0174 0.0161 0.0149 0.0138
3 0.0767 0.0719 0.0674 0.0632 0.0591 0.0554 0.0518 0.0485 0.0453 0.0424
4 0.1641 0.1555 0.1473 0.1395 0.1321 0.1249 0.1181 0.1117 0.1055 0.0996
5 0.2881 0.2759 0.2640 0.2526 0.2414 0.2307 0.2203 0.2103 0.2006 0.1912
6 0.4349 0.4204 0.4060 0.3920 0.3782 0.3646 0.3514 0.3384 0.3257 0.3134
Continue;
34

λ
x 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0
7 0.5838 0.5689 0.5541 0.5393 0.5246 0.5100 0.4956 0.4812 0.4670 0.4530
8 0.7160 0.7027 0.6892 0.6757 0.6620 0.6482 0.6343 0.6204 0.6065 0.5925
9 0.8202 0.8096 0.7988 0.7877 0.7764 0.7649 0.7531 0.7411 0.7290 0.7166
10 0.8942 0.8867 0.8788 0.8707 0.8622 0.8535 0.8445 0.8352 0.8257 0.8159
11 0.9420 0.9371 0.9319 0.9265 0.9208 0.9148 0.9085 0.9020 0.8952 0.8881
12 0.9703 0.9673 0.9642 0.9609 0.9573 0.9536 0.9496 0.9454 0.9409 0.9362
13 0.9857 0.9841 0.9824 0.9805 0.9784 0.9762 0.9739 0.9714 0.9687 0.9658
14 0.9935 0.9927 0.9918 0.9908 0.9897 0.9886 0.9873 0.9859 0.9844 0.9827
15 0.9972 0.9969 0.9964 0.9959 0.9954 0.9948 0.9941 0.9934 0.9926 0.9918
16 0.9989 0.9987 0.9985 0.9983 0.9980 0.9978 0.9974 0.9971 0.9967 0.9963
17 0.9996 0.9995 0.9994 0.9993 0.9992 0.9991 0.9989 0.9988 0.9986 0.9984
18 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994 0.9993
19 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997
20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999
21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0
0 0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 0.0001
1 0.0028 0.0025 0.0023 0.0021 0.0019 0.0018 0.0016 0.0015 0.0014 0.0012
2 0.0127 0.0118 0.0109 0.0100 0.0093 0.0086 0.0079 0.0073 0.0068 0.0062
3 0.0396 0.0370 0.0346 0.0323 0.0301 0.0281 0.0262 0.0244 0.0228 0.0212
4 0.0940 0.0887 0.0837 0.0789 0.0744 0.0701 0.0660 0.0621 0.0584 0.0550
5 0.1822 0.1736 0.1653 0.1573 0.1496 0.1422 0.1352 0.1284 0.1219 0.1157
6 0.3013 0.2896 0.2781 0.2670 0.2562 0.2457 0.2355 0.2256 0.2160 0.2068
7 0.4391 0.4254 0.4119 0.3987 0.3856 0.3728 0.3602 0.3478 0.3357 0.3239
8 0.5786 0.5647 0.5507 0.5369 0.5231 0.5094 0.4958 0.4823 0.4689 0.4557
9 0.7041 0.6915 0.6788 0.6659 0.6530 0.6400 0.6269 0.6137 0.6006 0.5874
10 0.8058 0.7955 0.7850 0.7743 0.7634 0.7522 0.7409 0.7294 0.7178 0.7060
11 0.8807 0.8731 0.8652 0.8571 0.8487 0.8400 0.8311 0.8220 0.8126 0.8030
12 0.9313 0.9261 0.9207 0.9150 0.9091 0.9029 0.8965 0.8898 0.8829 0.8758
13 0.9628 0.9595 0.9561 0.9524 0.9486 0.9445 0.9403 0.9358 0.9311 0.9261
14 0.9810 0.9791 0.9771 0.9749 0.9726 0.9701 0.9675 0.9647 0.9617 0.9585
15 0.9908 0.9898 0.9887 0.9875 0.9862 0.9848 0.9832 0.9816 0.9798 0.9780
16 0.9958 0.9953 0.9947 0.9941 0.9934 0.9926 0.9918 0.9909 0.9899 0.9889
17 0.9982 0.9979 0.9977 0.9973 0.9970 0.9966 0.9962 0.9957 0.9952 0.9947
18 0.9992 0.9991 0.9990 0.9989 0.9987 0.9985 0.9983 0.9981 0.9978 0.9976
19 0.9997 0.9997 0.9996 0.9995 0.9995 0.9994 0.9993 0.9992 0.9991 0.9989
20 0.9999 0.9999 0.9998 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996
21 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998
22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999
23 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
35

x 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0
0 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0000
1 0.0011 0.0010 0.0009 0.0009 0.0008 0.0007 0.0007 0.0006 0.0005 0.0005
2 0.0058 0.0053 0.0049 0.0045 0.0042 0.0038 0.0035 0.0033 0.0030 0.0028
3 0.0198 0.0184 0.0172 0.0160 0.0149 0.0138 0.0129 0.0120 0.0111 0.0103
4 0.0517 0.0486 0.0456 0.0429 0.0403 0.0378 0.0355 0.0333 0.0312 0.0293
5 0.1098 0.1041 0.0986 0.0935 0.0885 0.0838 0.0793 0.0750 0.0710 0.0671
6 0.1978 0.1892 0.1808 0.1727 0.1649 0.1574 0.1502 0.1433 0.1366 0.1301
7 0.3123 0.3010 0.2900 0.2792 0.2687 0.2584 0.2485 0.2388 0.2294 0.2202
8 0.4426 0.4296 0.4168 0.4042 0.3918 0.3796 0.3676 0.3558 0.3442 0.3328
9 0.5742 0.5611 0.5479 0.5349 0.5218 0.5089 0.4960 0.4832 0.4705 0.4579
10 0.6941 0.6820 0.6699 0.6576 0.6453 0.6329 0.6205 0.6080 0.5955 0.5830
11 0.7932 0.7832 0.7730 0.7626 0.7520 0.7412 0.7303 0.7193 0.7081 0.6968
12 0.8684 0.8607 0.8529 0.8448 0.8364 0.8279 0.8191 0.8101 0.8009 0.7916
13 0.9210 0.9156 0.9100 0.9042 0.8981 0.8919 0.8853 0.8786 0.8716 0.8645
14 0.9552 0.9517 0.9480 0.9441 0.9400 0.9357 0.9312 0.9265 0.9216 0.9165
15 0.9760 0.9738 0.9715 0.9691 0.9665 0.9638 0.9609 0.9579 0.9546 0.9513
16 0.9878 0.9865 0.9852 0.9838 0.9823 0.9806 0.9789 0.9770 0.9751 0.9730
17 0.9941 0.9934 0.9927 0.9919 0.9911 0.9902 0.9892 0.9881 0.9870 0.9857
18 0.9973 0.9969 0.9966 0.9962 0.9957 0.9952 0.9947 0.9941 0.9935 0.9928
19 0.9988 0.9986 0.9985 0.9983 0.9980 0.9978 0.9975 0.9972 0.9969 0.9965
20 0.9995 0.9994 0.9993 0.9992 0.9991 0.9990 0.9989 0.9987 0.9986 0.9984
21 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9995 0.9994 0.9993
22 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9997
23 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
24 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0
0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
2 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
3 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000
4 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001 0.0000 0.0000
5 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003 0.0002 0.0001
6 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010 0.0005 0.0003
7 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029 0.0015 0.0008
8 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071 0.0039 0.0021
9 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154 0.0089 0.0050
10 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304 0.0183 0.0108
11 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549 0.0347 0.0214
12 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917 0.0606 0.0390
13 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426 0.0984 0.0661
14 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081 0.1497 0.1049
15 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867 0.2148 0.1565
36

Continue;
λ
x 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0
16 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751 0.2920 0.2211
17 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686 0.3784 0.2970
18 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622 0.4695 0.3814
19 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509 0.5606 0.4703
20 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307 0.6472 0.5591
21 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991 0.7255 0.6437
22 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551 0.7931 0.7206
23 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989 0.8490 0.7875
24 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317 0.8933 0.8432
25 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554 0.9269 0.8878
26 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718 0.9514 0.9221
27 1.0000 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827 0.9687 0.9475
28 1.0000 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897 0.9805 0.9657
29 1.0000 1.0000 1.0000 0.9999 0.9996 0.9989 0.9973 0.9941 0.9882 0.9782
30 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9986 0.9967 0.9930 0.9865
31 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993 0.9982 0.9960 0.9919
32 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9990 0.9978 0.9953
33 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9988 0.9973
34 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9985
35 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9992
36 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996
37 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998
38 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
39 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
40 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
37

STANDARD NORMAL DISTRIBUTION
2
2
1
( )
2
z x
P Z z e dx
π
−
−∞
≤ = ∫
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Standard Normal Distribution Table
0 z
38

Continue;
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
-0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
-3.6 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
39

ANSWERS
EXERCISE 1
1. p = 0.05, q = 1 – 0.05 = 0.95, n = 6
a) P(X=3) = 6
C3(0.05)3
(0.95)6-3
= 0.0021
b) P(X < 2) = P(X = 0) + P(X = 1)
= 6
C0(0.05)0
(0.95)6-0
+
6
C1(0.05)1
(0.95)6-1
= 0.7351 +0.2321
= 0.9672
c) P(X=0) = 6
C0(0.05)0
(0.95)6-0
= 0.7351
2. p = 0.30, q = 1 – 0.30 = 0.70, n = 5
P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X=5)
= 5
C3 (0.30)3
(0.70)2
+
5
C4 (0.30)4
(0.70)1
+
5
C5 (0.30)5
(0.70)0
= 0.1323 + 0.0283 + 0.0024
= 0.1631
3. p = 0.40, q = 1 – 0.40 = 0.60, n = 5
a) P(X=2) = 5
C2(0.04)2
(0.6)5-2
= 0.3456
b) P(X ≤ 3) = 1- ( P(X = 4) + P(X = 5))
= 1 – (5
C4 (0.4)4
(0.6)1
+
5
C5 (0.4)5
(0.6)0
)
= 1 – (0.0768 +0.01024
= 0.91296
c) P( X ≥ 2 ) = P(X = 0) + P(X=1) + ( P=2)
= 5
C0 (0.4)0
(0.6)5
+
5
C1 (0.4)1
(0.6)4
+ 5
C2 (0.4)2
(0.6)3
= 0.07776 + 0.2592 + 0.3456
= 0.6826
4. p = 0.6, q = 1 – 0.6 = 0.4, n = 10
P(X=3) = 10
C3 (0.6)3
(0.4)7
= 0.042
5. p = 0.65, q = 0.35, n = 10
a) P(X = 4) = 10
C4 (0.65)4
(0.35)10-4
= 0.0689
p = 0.35, q = 0.65, n = 10
b) P(X ≥ 6) = 1 – P(X ≤ 5)
= 1 – 0.9051
= 0.0949
6. p = 0.4, q = 0.6, n = 15
a) P(X ≤ 5) = 0.4032
b) P(X < 4) = P(X ≤ 3)
= 0.0905
c) P(X ≥ 9) = 1 – P(X ≤ 8)
= 1 – 0.9050
= 0.095
d) P(6 ≤ X ≤ 9) = P(X ≤ 9) – P(X ≤ 5)
= 0.9662 – 0.4032
= 0.5630
e) P(4 < X < 8) = P( X ≤ 7) – P(X ≤ 4)
= 0.7869 – 0.2173
= 0.5696
EXERCISE 2
1. µ = np = 200(0.83) = 166
σ2
= npq = 200(0.83)(0.17) = 28.22
σ = √npq = √28.22 = 5.3122
2. µ = np = 1000(0.25) = 250
σ2
= npq = 1000(0.25)(0.75) = 187.5
σ = √npq = √187.5 = 13.6931
40

EXERCISE 3
1. λ = 2 telemarketing phone calls per week
a) P(X = 6) =
2 6
2
6!
e−
= 0.0120
λ = 8 telemarketing phone calls in one month
b) P(X < 3) = P(X ≤ 2)
= P(X = 0) + P(X = 1) +
P(X = 2)
=
8 0 8 1 8 2
8 8 8
0! 1! 2!
e e e− − −
+ +
= 0.0003 + 0.0027 + 0.0107
= 0.0137
2. λ = 3 times breakdown per month
a)
3 2
3
P(X=2)=
2!
e−
= 0.2240
b) P(X ≤ 1) =
3 0 3 1
3 3
0! 1!
e e− −
+
= 0.0498 + 0.1494
= 0.1992
λ = 6 times breakdown in 2 months
c) P(X = 0) =
6 0
6
0!
e−
= 0.0025
3. λ = 0.9 plain lands at the airport per 5 min interval.
a) P(X ≤ 1) = P(X=0) + P(X=1)
= (e-0.9
(0.9)0
)/0! + (e-0.9
(0.9)1
)/1!
= 0.4066 + 0.3659
= 0.772
λ = 0.9*3 = 2.7 plains land at the airport per 15 min
interval.
b) P(X > 3) = 1- P (X ≤ 3)
= 1 – ((P(X=0) + P(X=1) + P(X=2) +
P(X=3)
= 1 – ((e-2.7
(2.7)0
)/0! +
(e-2.7
(2.7)1
)/1!)/1! +
(e-2.7
(2.7)2
)/2!) + (e-2.7
(2.7)3
)/3!)
= 1 – (0.0672 + 0.1815 + 0.245 +
0.2205)
= 1 – 0.7142
= 0.286
4. λ = 4.8 customer every half hour
a) P(X = 2) = P(X ≤ 2) – P(X ≤ 1)
= 0.1425 – 0.0477
= 0.0948
b) P(X ≤ 2) = 0.1425
c) P(X = 0) = P(X ≤ 0) = 0.0082
λ = 9.6 customer in one hour
d) P(X > 5) = 1 – P(X ≤ 5)
= 1 – 0.0838
= 0.9162
5. λ = 8 calls per hour
a) P (X ≥ 8) = 1 – P(X ≤ 7)
= 1 – 0.4530
= 0.547
b) P (X ≥ 3) = 1 - P(X ≤ 3)
= 1 – 0.0138
= 0.9862
c) P (X ≤ 5) = 0.1912
41

42

EXERCISE 4
a) To the left of z = -2.87
P(z < -2.87) = 0.0021
b) To the right of z = 2.45
P(z > 2.45) = 1 - P(z < 2.45)
= 1 - 0.9929
= 0.0071
c) Between z = -2.15 and z = 1.67
P(-2.15 < z < 1.67)
= P(z < 1.67) – P(z < -2.15)
= 0.9525 – 0.0158 = 0.9367
EXERCISE 5
1. a) P(X < 150) = P(z < 150 - 190)
21
= P(z < -1.9)
= 0.0281
b) P(205 < X < 245)
= P(z < 2.62) - P(z < 0.71)
= 0.9956 – 0.7611
= 0.2345
2. P(X < 3.5) = P(z < 3.5 – 3.1)
0.5
= P(z < 0.8)
= 1 - 0.7881
= 0.2119
= 21.19%
3. n = 6000, μ = 55, σ = 10
a) P(X > 75) = P(z < 75 – 55)
10
= P(z < 2)
= 1 - P(z < 2)
= 1 - 0.9772
= 0.0228
Thus, the number of graduates with distinction
is
0.0228( 6000) = 136.8
≈ 137 graduates
b) P(65 < X < 45) = P[(65 – 55) <z < 45 – 55)]
10 10
= P( 1 < z < -1)
= 0.8413 – 0.1587
= 0.6826
-2.87 0
0 2.45
-2.15 1.67

EXERCISE 6
1. p = 0.1, q = 0.9, n = 5
a) P(X=2) = 5
C2(0.1)2
(0.9)3
= 0.0729
or
P(X=2) = P(X≤2) – P(X≤1)
= 0.9914 – 0.9188
= 0.0729
b) P (X ≥ 1) = 1 – P(X=0)
= 1 – [5
C0(0.1)0
(0.9)5
]
= 1 – 0.5905
= 0.4095
or
P(X≥ 1) = 1 – P(X≤0)
= 1 – 0.5905
= 0.4095
2. p = 0.2, q = 0.8, n = 20
a) P(X < 3) = P(X≤2)
= 0.2061
or
P(X < 3) = P(X=0) + P(X=1) + P(X=2)
= 20
C0(0.2)0
(0.8)20
+ 20
C1(0.2)1
(0.8)19
+
20
C2(0.2)2
(0.8)18
= 0.01152 + 0.0576 + 0.1369
= 0.2061
b) P(X=4) = 20
C4(0.2)4
(0.8)16
= 0.2182
or
P(X=4) = P(X≤4) - P(X≤3)
= 6296 – 0.4114
= 0.2061
3. p = 0.4, q = 0.6, n =20
P(X>10) = 1 – P(X≤9)
= 1 – 0.4044
= 0.5956
4. p = 0.12, q = 0.88, n = 75
µ = np = 75(0.12) = 9
σ2
= npq = 75(0.12)(0.88) = 7.92
σ = √npq = √7.92 = 2.814
5. p = 0.88, q = 0.12, n = 16
a) P(X=9) = 16
C9(0.88)9
(0.12)7
= 0.0013
p = 0.08, q = 0.92, n = 16
b) P(X=4) = 16
C4(0.08)4
(0.92)12
= 0.0274
p = 0.04, q = 0.96, n = 16
c) P(X=3) = 16
C3(0.04)3
(0.96)13
= 0.0211
6. λ = 3 loan applications per week
a) P(X = 0) = (e-3
(3)0
)/0!
= 0.0498
b) P(2 ≤ z ≤ 5) = P(X≤5) - P(X≤2)
= 0.9161 – 0.4232
= 0.4929
λ = 6 loan applications per 2 weeks
c) P (X ≥ 5) = 1 - P(X ≤ 4)
= 1 – 0.2851
= 0.7149
7. λ = 5 books from each lot
P(X = 5) = (e-5
(5)5
)/5!
= 0.1754
8. λ = 42/60 = 0.7 customer per minute
a) P(X = 0) = (e-0.7
(0.7)0
)/0!
= 0.4966
λ = 0.7(5) = 3.5 customer per five minutes
b) P (X ≥ 4) = 1 - P(X ≤ 3)
= 1 – 0.5366
= 0.4634
λ = 0.7(10) = 3.5 customer per ten minutes
c) P (2 < X < 6) = P(X ≤ 5) - P(X ≤ 2)
= 0.8576 – 0.3208
= 0.5368
9. μ = 2.8, σ = 4
a) P(2 < X < 3) =P(X<3) - P(X<2)
=P[((3 – 2.8)/4) <z < ((2 – 2.8)/4]
= P(0.05 < z < -0.2)
= 0.8413 – 0.4207
= 0.4206

b) P(X<2) = P[z < (2 - 2.8)/4]
= P (z < -2)
= 0.0228
c) P(X >3) = 1 - P(X<3)
= 1 – P [z < (3 - 2.8)/4]
= 1 – P (z < 0.05)
= 1 – 0.5199
= 0.4801
d) P(X >3.7) = 1 - P(X<3.7)
= 1 – P [z < (3.7 – 2.8)/4]
= 1 – P (z < 0.23)
= 1 – 0.5910
= 0.4090
Thus, the no. of 1st
class honors student is:
0.4090(1000) = 409 1st
class honors students
10. μ = 24, σ = 3.8
a) P(X >30) = 1 - P(X<30)
= 1 – P [z < (30 – 24)/3.8]
= 1 – P (z < 1.58)
= 1 – 0.9394
= 0.0606
b) P(20 < X < 25) =P(X≤25) - P(X≤20)
=P[((25 – 24)/3.8) <z < ((20 – 24)/3.8]
= P( 0.26 < z < -1.05)
= 0.6026 – 0.1469
= 0.4557
c) P(X >25) = 1 - P(X<25)
= 1 – P [z < (25 – 24)/3.8]
= 1 – P (z < 0.26)
= 1 – 0.6026
= 0.3974
d) P(X<10) = P[z < (2 - 2.8)/04]
= P (z < -2)
= 0.0228
11. μ = 15, σ = 3
a) P(X<13) = P[z < (13 - 15)/3]
= P (z < -0.67)
= 0.2514
b) P(X >16) = 1 - P(X<16)
= 1 – P [z < (16 – 15)/3]
= 1 – P (z < 0.33)
= 1 – 0.6293
= 0.3707
c) P(14 < X < 17) =P(X<17) - P(X<14)
=P[((17 – 15)/3) <z < ((14 – 15)/3]
= P(0.67 < z < -0.33)
= 0.7486 – 0.3707
= 0.3779
12. μ = 132, σ = 8
a) P(X >130) = 1 - P(X<130)
= 1 – P [z < (130 – 132)/8]
= 1 – P (z < -0.25)
= 1 – 0.4013
= 0.5987
b) P(X<140) = P[z < (140 - 132)/8]
= P (z < 1.00)
= 0.8413
c) P(131 < X < 136) =P(X<136) - P(X<131)
=P[((136 – 132)/8) <z < ((131 – 132)/8]
= P(0.5 < z < -0.13)
= 0.6915 – 0.4483
= 0.2432
13. μ = 80, σ = 12
a) P(X<90) = P[z < (90 - 82)/12]
= P (z < 0.67)
= 0.7486
b) P(X >75) = 1 - P(X<75)
= 1 – P [z < (75 – 80)/12]
= 1 – P (z < -0.42)
= 1 – 0.3372
= 0.6628
c) P(79 < X < 95) =P(X<95) - P(X<79)
=P[(95 – 80)/12) <z < (79 – 80)/12]
= P(1.25 < z < -0.08)
= 0.8944 – 0.4681
= 0.4263

Matrix No: ____________________ Group: __________
TUTORIAL CHAPTER 5
QUESTION 1
A past history shows that 60% of college students are smokers. A sample of 5 students is randomly selected.
What is the probability that exactly one student is a smoker?
(2 marks)
QUESTION 2
Hope.fm Radio Company receives, on the average, two short messaging services (sms) within five minutes
from listener. Find the probability that it will receive the following
i. at least four sms within five minutes
(2 marks)
ii. two sms within fifteen minutes
(3 marks)
Tutorial
47
14

QUESTION 3
EJ Utara Transport determined that on an annual basis, the distance travel per lorry is normally distributed,
with a mean of 50 thousand kilometers and a standard deviation of 10 thousand kilometers. Find the
i. probability that a randomly selected lorry travels from 36 to 50 thousand kilometers in the year
(3 marks)
ii. minimum distance traveled by 80% of the lorries.
(4 marks)
Tutorial
48

Elementary Statistics Special Distributions

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