1. Stat310 The normal distribution
Hadley Wickham
Thursday, 12 March 2009

2. 1. Exam
2. Recap
3. Finish off convergence example
4. The normal distribution (reading?)
Thursday, 12 March 2009

3. Exam
Graded for you to pick up after class.
Generally did ok on question one, despite the
mistake. Point for question four just for
attempting it
Question was fine too.
Struggled with question three (which was
supposed to be pretty straightforward - sorry!)
“Carry through”
Thursday, 12 March 2009

4. Question 3
Thursday, 12 March 2009

5. 1.0 ● ● ●
● ●
●
● ● ● ●●
● ● ●●
● ●
● ● ●
●
● ● ●
● ●●
0.8 ● ● ● ● ●
●
● ●
● ●
●
●
● ●
0.6 ●
●
●
exam2/30
● ● ● ●
●
●
●
0.4 ●
0.2
0.0 ●
0.2 0.4 0.6 0.8 1.0
exam1/50
Thursday, 12 March 2009

6. 8
6
count
4
2
0
0.0 0.2 0.4 0.6 0.8 1.0
exam2/30
Thursday, 12 March 2009

7. Recap
If X1, X2, …, Xn are iid, then:
What does the joint pdf look like?
What is the expected value of the sum?
What is the variance of the sum?
What is the mgf of the sum?
Thursday, 12 March 2009

8. Convergence in P
Imagine you have a Bernoulli(p) process.
You can repeat the process as many
times as you like to generate X1, X2, …,
Xn. How could you use these X’s to figure
out what p is?
Thursday, 12 March 2009

9. lim P (|Zn − p| ≤ ) = 1
n→∞
∀ >0
Thursday, 12 March 2009

10. Time Event Estimate
Total
1 1 1 1.00
2 0 1 0.50
3 1 2 0.67
4 0 2 0.50
5 0 2 0.40
Thursday, 12 March 2009

11. 1.0
0.8
0.6
est
0.4
0.2
0.0
200 400 600 800 1000
n
Thursday, 12 March 2009

12. 1.0
0.8
0.6
est
0.4
0.2
0.0
200 400 600 800 1000
n
Thursday, 12 March 2009

13. 1.0
0.8
0.6
est
0.4
0.2
0.0
200 400 600 800 1000
n
Thursday, 12 March 2009

14. 1000 runs
Thursday, 12 March 2009

15. 1.0
0.8
99%
0.6
est
0.4
75%
25%
0.2
1%
0.0
200 400 600 800 1000
n
Thursday, 12 March 2009

16. 0.40
99%
0.35
75%
0.30
est
50%
25%
0.25
1%
0.20
200 400 600 800 1000
n
Thursday, 12 March 2009

17. 0.7
0.6
0.5
0.4
dist
0.3
0.2
0.1
0.0
200 400 600 800 1000
n
Thursday, 12 March 2009

18. 0.10
99%
0.08
0.06
dist
75%
0.04
50%
0.02
25%
0.00
1%
200 400 600 800 1000
n
Thursday, 12 March 2009

19. The normal
distribution
Thursday, 12 March 2009

20. 1 2
(x−µ)
− 2σ2
f (x) = √ e
2π
Is this a valid pdf?
Thursday, 12 March 2009

21. 1 22
M (t) = e µt+ 2 σ t
See book for derivation.
A few tricks + lots of algebra
Thursday, 12 March 2009

22. Your turn
σ
If X ~ Normal(μ, 2), what is the mean and
variance of X?
(Work it out - don’t just write down what
you know!)
Thursday, 12 March 2009

23. 0.4 0.4
N(-2, 1) N(5, 1)
0.3 0.3
0.2 0.2
f(x)
f(x)
0.1 0.1
0.0 0.0
−10 −5 0 5 10 −10 −5 0 5 10
0.4
N(0, 1)
0.3
0.2
f(x)
0.1
0.0
−10 −5 0 5 10
0.4 0.4
N(0, 4) N(0, 16)
0.3 0.3
0.2 0.2
f(x)
f(x)
0.1 0.1
0.0 0.0
−10 −5 0 5 10 −10 −5 0 5 10
Thursday, 12 March 2009

24. Transformations
σ
If X ~ Normal(μ, 2), and Y = a(X + b)
Y ~ Normal(b + μ, 2σ2)
a
If a = -μ and b = 1/σ, we often write
Z = (X - μ) / σ
Z ~ Normal(0, 1) = standard normal
Thursday, 12 March 2009

25. Example
Let X ~ Normal(5, 10)
What is P(3 < X < 8) ?
Convert to standard normal. Look up Z
score
P(-0.2 < Z < 0.3) = P(Z < 0.3) - P(-0.2)
(Google z table)
Thursday, 12 March 2009

26. P (Z < z) = Φ(z)
Φ(−z) = 1 − Φ(z)
P (−1 < Z < 1) = 0.68
P (−2 < Z < 2) = 0.95
P (−3 < Z < 3) = 0.998
Thursday, 12 March 2009

27. Readings: 5.3, 5.4, 5.5
Thursday, 12 March 2009