QT1 - 05 - Probability Distributions

Probability Distributions Q U A N T T E C H I N T E U Q I A S E V I T 1 0 S

Contents ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Frequency Distributions (Again!) ,[object Object],[object Object],[object Object]

From Frequency Distributions to Probability Distributions ,[object Object],[object Object],[object Object],[object Object],[object Object]

Probability Distribution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Prediction ,[object Object],[object Object],[object Object],[object Object],[object Object]

Basic Modelling Technique Toss a Coin; Head or Tail How many Heads in 10 Coin Tosses Chest Size of Army Recruits Amount of Protein in an Egg Number of Members in a Family Money spent by a family at multiplex Number of Defects per thousand Percentage of People who die before 30 Experimental Data or observation Can be represented by a set of random numbers That obeys certain rules .. 1 or 0 Integer between 0 and 10 Integer between 30 and 50 Any number between 3 and 5 Integer between 0 and 10 Any positive number less than 2000 Integer between 0 and 5 Any number from 0 and 100

Random Variable ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Expected Value ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Bernoulli Random Variable ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],R=1 R=0 p q p+q=1 p Probability Cumulative probability

Physical Bernoulli Processes ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Another Bernoulli Event ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Another Bernoulli Scenario ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Characteristics of a Bernoulli Event ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Simulating a few Bernoulli Event ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Simulating Many Bernoulli Events

Bernoulli to Binomial ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Binomial Random Variable ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Binomial Distribution : Experiment and Theory

Practical Considerations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Determination of the p value – the Bernoulli success probability – is very important but equally difficult

Central Tendency & Dispersion of Binomial Distribution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Impact of 'p' on Binomial Probability

Impact of 'n' on Binomial Probability

Poisson Distribution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Poisson and Binomial ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Equivalence of Binomial & Poisson ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],The Poisson probability is a good approximation of the Binomial probability when the number of trials becomes very high Example : Events (“arrivals”) are happening every minute or second !

Bernoulli => Binomial => Poisson

Similarity of Processes ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Poisson Distribution in terms of average number of events ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Poisson @ childbirth ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Two Hospitals ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Poisson Distribution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Poisson Distribution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Exponential Distribution Interval between two Arrivals ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Close Cousins in Randomness ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

QT1 - 05 - Probability Distributions

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QT1 - 05 - Probability Distributions