1. Stat310 Sampling distributions
Hadley Wickham
Monday, 22 March 2010

2. Quiz
• Pick up quiz on your way in
• Start at 1pm
• Finish at 1:10pm
• Closed book
Monday, 22 March 2010

3. http://xkcd.com/715/
Monday, 22 March 2010

4. 1. Quiz
2. CLT & approximations
3. Sampling distributions
4. Example
5. More theory
Monday, 22 March 2010

5. CLT
Central limit theorem.
The distribution of a mean is normal when
gets big.
Monday, 22 March 2010

6. Approximation
This implies that if n is big then ...
Monday, 22 March 2010

7. Sampling
distributions
Monday, 22 March 2010

8. Random experiment
“A random experiment is an experiment,
trial, or observation that can be repeated
numerous times under the same conditions...
It must in no way be affected by any previous
outcome and cannot be predicted with
certainty.” (http://cnx.org/content/m13470/latest/)
i.e. it is uncertain (we don’t know ahead of
time what the answer will be) and repeatable
(ideally).
Monday, 22 March 2010

9. Where we are
Univariate random variables:
an experiment with one output
Bivariate random variables:
an experiment with two outputs
Sequences of random variables:
An experiment performed repeatedly.
Repeatable = i.i.d
Monday, 22 March 2010

10. A sampling distribution:
Summary statistics from
a repeated experiment
Monday, 22 March 2010

11. Definitions
Sample = results of n random experiments.
Random sample = result of a random
experimented repeated n times. Therefore,
they’re iid.
Both are sequences of random variables.
Statistic = A function of random variables
with no unknown parameters.
Monday, 22 March 2010

12. Example
Spin a bottle and record the angle in
degrees in which it points. Repeat.
How would you write this
mathematically?
Monday, 22 March 2010

13. First time
x1 = 205, x2 = 256, x3 = 86, x4 = 119,
x5 = 16, x6 = 278, x7 = 55, x8 = 16,
x9 = 295, x10 = 341, x11 = 299, x12 = 270,
x13 = 118, x14 = 360, x15 = 97, x16 = 282,
x17 = 42, x18 = 283, x19 = 259, x20 = 326
Monday, 22 March 2010

14. Second time
x1 = 184, x2 = 344, x3 = 118, x4 = 226,
x5 = 208, x6 = 106, x7 = 332, x8 = 310,
x9 = 339, x10 = 95, x11 = 7, x12 = 274,
x13 = 120, x14 = 346, x15 = 211, x16 = 166,
x17 = 84, x18 = 102, x19 = 32, x20 = 128
Monday, 22 March 2010

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Value
Monday, 22 March 2010

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Value
Monday, 22 March 2010

17. 4
3
count
2
1
0
140 160 180 200
samp
Monday, 22 March 2010

18. 8000
6000
count
4000
2000
0
100 150 200 250
V1
Monday, 22 March 2010

19. What will happen as I
8000
vary the number of
samples I average
over? (What theorem
6000 applies here?)
count
4000
2000
0
100 150 200 250
V1
Monday, 22 March 2010

20. 1 2 3
400
300
200
100
0
count
4 5
400
300
200
100
0
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
mean
Monday, 22 March 2010

21. 1 10
4000
3000
2000
1000
0
count
100 1000
4000
3000
2000
1000
0
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
mean
Monday, 22 March 2010

22. 1 10
How can I transform
4000
this random variable to
3000 make it comparable?
(What theorem applies
2000
here?)
1000
0
count
100 1000
4000
3000
2000
1000
0
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
mean
Monday, 22 March 2010

23. 1 2 3 4
800
600
400
200
0
5 10 20 100
800
600
count
400
200
0
1000 10000
800
600
400
200
0
−400 −200 0 200 400 −400 −200 0 200 400 −400 −200 0 200 400 −400 −200 0 200 400
(mean − 180) * sqrt(n)
Monday, 22 March 2010

24. 2 3
We can do the same thing
1000
for other statistics...
800
600
400
200
0
count
4 5
1000
800
600
400
200
0
0 50 100 150 200 250 0 50 100 150 200 250
sqrt(var)
Monday, 22 March 2010

25. 2 3 4
600 700
500 600
500
400 500
400
300 400
300 300
200 200
200
100 100 100
0 0 0
0 50 100 150 200 250 0 50 100 150 200 0 50 100 150
5 10 20
1000
600 800 800
600
count
600
400
400 400
200 200 200
0 0 0
50 100 150 40 60 80 100 120 140 160 60 80 100 120 140
100 1000 10000
800 1000
800
800
600 600
600
400 400
400
200 200 200
0 0 0
90 95 100 105 110 115 120 98 100 102 104 106 108 110 102.503.003.504.004.505.005.5
1 1 1 1 1 1
sqrt(var)
Monday, 22 March 2010

26. Theory
We’ll start with the mean of normally
distributed random variables, then try to
extend in various ways.
Monday, 22 March 2010

27. Your turn
X1, X2, ... are iid N(μ, σ2)
n
¯ Sn
Sn = Xi Xn =
n
1
Find their mgfs. What do you notice?
Hint: MX (t) = exp µt + σ t 2 2
Monday, 22 March 2010

28. Reading
4.2, 4.2.1
4.2.2, 4.4
Monday, 22 March 2010