1.
Stat310 Normal distribution
Hadley Wickham
Thursday, 19 March 2009

2.
1. Another summer opportunity
2. Recap
3. Standard normal
4. Sums of normals
5. Chi-square distribution
Thursday, 19 March 2009

3.
VIGRE research project
Over summer
$5,000
Work with me (or anyone else in stats
department)
Email me if you’re interested
Thursday, 19 March 2009

4.
Recap
What is the pdf of the normal distribution?
What is the mgf?
What is the mean and variance?
How do you create a standard normal?
What is the pdf and mgf of the gamma
distribution?
Thursday, 19 March 2009

5.
Standard normal
If X ~ Normal(μ, σ2), and
Z = (X - μ) / σ
Then:
Z ~ Normal(0, 1) = standard normal
How can we show this?
(What if X isn’t normal?)
Thursday, 19 March 2009

6.
Using the
standard normal
These days, you don’t need to use the
standard normal, you can just use a
computer.
But it’s useful for exams, and more
importantly, it’s how statisticians tend to
think about the normal distribution
Use plot to give rough estimate.
Thursday, 19 March 2009

7.
Using the tables
Column + row = z
Find: Φ(2.94), Φ(-1), Φ(0.01), Φ(4)
Can also use in reverse: For what value
of z is P(Z < z) = 0.90 ? i.e. What is Φ-1(0.90)?
Find: Φ-1(0.1), Φ-1(0.5), Φ-1(0.65), Φ-1(1)
Thursday, 19 March 2009

8.
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, 19 March 2009

9.
Example
The time it takes me to bike to school is
normally distributed with mean 10 and
standard deviation 4.
What is the probability it takes me more
than 20 minutes to bike to school?
What time should I leave so that I have
95% chance of getting to class by 1pm?
Thursday, 19 March 2009

10.
Example
What’s the probability I take a negative
amount of time to get to school?
Is the distribution of my bike times really
normal?
Thursday, 19 March 2009

11.
Confidence interval
I’d like to create a 95% confidence
interval for my biking time. i.e. I want to
find a and b such that P(a < X < b) = 0.95.
How many ways are there to construct
this interval?
Generally want to find the interval with the
shortest length. How can I do that?
Thursday, 19 March 2009

12.
Sums of normals
Normal(μi, σi
Let Xi ~ 2), independent
Y = c1X1 + c2X2 + … + cnXn
What is the distribution of Y?
(What is the mean and variance of Y?)
How can we work it out?
Thursday, 19 March 2009

13.
Example
If Z1, Z2, Z3 are independent standard
normals, what is the distribution of:
Z1 - Z3
Z1 + Z2 + Z3
(Z1 + Z2 + Z3)/3
Thursday, 19 March 2009

14.
CLT prequel
If X1, X2, …, Xn are iid N(μ, σ2)
What is the distribution of their average?
Thursday, 19 March 2009

15.
Another special
distribution
If X ~ Normal(μ, σ2), and
V = (X - μ)2 / σ2 = Z2
Then
V ~ χ2(1)
Thursday, 19 March 2009

16.
Chi-squared
Skipped over it in Chapter 3
Special case of the gamma distribution,
when θ = 2 and α = r / 2 (r an integer)
Mean = r, Variance = 2r
r is called degrees of freedom
Thursday, 19 March 2009

17.
http://en.wikipedia.org/wiki/Chi-square_distribution
Thursday, 19 March 2009

18.
Sums
Let Z1, Z2, …, Zn be iid N(0, 1)
W = Z12 + Z22 + … + Zn2
Then
W ~ χ2(n)
This is going to be useful when we try to
estimate the variance
Thursday, 19 March 2009

19.
Why?
Thursday, 19 March 2009

20.
20
15
●
●
● ●
●
●
time
10
●
●
●
●
5
0
1 2 3 4 5
day
Thursday, 19 March 2009

21.
20
15
●
●
● ●
●
●
●
●
●
● ●
time
10
●
●
●
●
5
0
1 2 3 4 5
day
Thursday, 19 March 2009

22.
20 ●
●
●
●
● ●
● ●
●
●
● ● ●
● ● ●
●
● ●
● ●
● ●
● ● ●
●●
● ●
● ●
●
● ● ●●
● ●
● ●●●
● ●
● ●
●
●●
●
●●
●
● ● ●
●
● ●
● ● ●●●
●
●● ● ●●
●
●● ● ●
●● ●
●●
●●
● ● ● ● ●●
●
●
● ● ●●
●●
●
●
●
● ●
●
●●● ●
● ●
●● ●
●
●● ● ● ● ●
● ●
● ●
●● ● ● ●●
●
15 ● ● ●
● ●● ●
● ●
●● ●● ●●
●
● ●
●●
●● ●
● ●
●● ●
●● ●
● ●
● ●● ●●● ● ●
●
●
● ●● ●
●
●● ●● ● ●
● ●● ● ●●
● ●
●
●● ●● ●
● ●
●● ●
●
●● ●
● ●●
● ●● ● ●● ● ●
● ● ●● ● ●
●
● ● ● ●● ● ●
●
●
●● ●● ●
●●
●
● ●●● ● ●
● ● ●●
●
●
●● ● ● ● ●●●●
●●
● ●
●
● ●
●
● ● ● ●● ●
●
●● ● ● ●●●
● ●
●● ●
●●
● ●
●● ● ● ●● ● ●●● ●
●● ●
● ●
●
● ● ●●
●
● ●
● ●● ● ●● ● ● ●
● ●●
●
● ●
● ● ●● ●● ● ● ● ● ● ●● ● ●
●●●● ● ● ● ● ●● ● ● ●
●
● ●
●
● ● ● ●● ● ● ● ●● ● ●
● ●● ●
●
●●●●
●
● ●● ●
● ●
●●
● ●● ● ● ● ●
●
●● ●
● ●●
●● ●
●
● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●
●●● ● ● ●● ●●●● ●●● ● ● ● ●
●
● ●● ●●● ●● ●
● ●
● ●
● ●● ●● ● ●● ● ●●
●
●● ●
● ●●
● ● ●●●
● ● ●●
●●
●● ● ●
● ●●
●
● ●
● ●
●
● ● ● ● ● ● ●● ●●● ●● ●●●● ● ●● ● ●●● ● ● ● ● ● ● ● ●●● ●● ●● ● ● ● ● ● ●● ●● ●
●
●● ●● ●● ●
●
●● ●●
●●
● ●
● ●● ● ●
● ●● ● ●● ●
● ● ●
●
●● ●
● ●●
● ● ●● ●● ● ● ●
● ● ●
●
● ●
● ● ● ●● ●
●● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●
●
●
● ●●
●
● ●
● ●
● ● ● ●●
●● ●
● ●● ●●
●● ●●
●
●● ●
● ● ●
● ●
● ●
●
●
●
● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ● ●●
●● ●
●
●
● ●●●●●●● ● ● ● ●
●
●● ●
● ● ●
● ● ●● ● ● ●●
●● ●
●● ● ●
● ●●●
● ●● ●● ● ● ●
●● ● ● ● ● ● ●
●
●
●● ●●
time
●
10 ●● ● ●
● ●● ●● ● ●● ●
●●
●●● ●●
● ●● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●●● ● ● ●●
●●
●● ●●
● ● ●●● ●
● ● ●●
● ●● ● ● ●
●
● ●
● ●
●●● ●
●●●
● ● ● ● ● ● ●●
●
● ●● ●
● ●
● ● ● ● ●● ● ● ●
●●
●● ● ● ● ●
● ●
● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ●
● ●
● ● ●●
●
●●
●● ●● ● ● ● ●●
●● ●●
●●
● ●●
●
● ● ● ●● ● ● ●
●● ●
●●
● ●● ● ●● ● ● ●●
●
●●
●
●● ●●●
●
●● ● ●
●● ● ● ● ● ● ●● ●
● ● ●● ●●
●● ●
● ●●
● ●● ●●●● ● ● ● ● ● ●●
● ● ● ●● ● ●
●
● ●●
● ●● ●●● ●●●
●
●
● ● ●● ● ●
● ●
●● ● ●●●
● ●
● ●● ●●
●
● ●
●● ●●
●● ● ● ●● ●● ●
● ●
●●
● ●●
●
●● ● ● ● ●● ●●●●● ●●●● ● ● ●
●●
● ● ● ●●
●
●
●● ●● ● ● ●● ● ●● ● ●●● ●●
● ● ●
●
● ●●
●● ●● ●●
●● ● ● ● ●● ●
●●
● ●●
●●
● ●
●●
●●
● ● ●●
●● ●● ● ● ●● ● ● ● ●
●● ●● ●
●● ● ● ● ●● ● ● ● ● ●●
● ●
● ● ●
●
● ●● ● ●
● ● ●
●● ● ●
●● ●● ● ● ●
● ●● ●
● ●● ● ● ●● ● ●● ●●
● ●● ●
●●● ● ●
● ●● ● ● ●
●●
● ●
●●
●
● ●
● ● ● ●
● ●● ●
●●● ●
●●
● ●
●● ●●
● ●
●
● ● ● ● ●●
●● ●
●● ●
●● ● ● ● ● ● ●●
● ●●
● ● ● ●
●●
●● ●
●
● ●
● ●
●● ●
● ● ●●
● ● ●●
●
● ●
● ●
● ● ●
●●
● ●
●
●
● ●
●●
●●
● ●
● ● ●
5 ● ● ●● ●
●
● ● ●
●
● ● ●●
●
●
●
● ●● ● ● ●● ●
●● ●
●● ●
●
● ●
●●
●● ●
●● ●
●● ●
● ●
●
● ● ●
●
● ●
●●● ● ● ●
●
●●
● ●
● ●
●● ●●
● ●
● ●
● ● ●
● ● ●
● ● ●
● ●
●●
● ●
●
● ●
● ●●
●
● ●
●
● ●
● ●
●
●
●
● ●
● ●
●
●
●
●●
●
●
●
●
●
0 ●
●
100 200 300 400 500
day
Thursday, 19 March 2009

23.
20
●
●
●
●
●
●
●
●
●
● ●
●
●
●● ●
●
●
● ●
● ●
●
15 ● ●
● ●
●
● ●
● ● ●
● ● ●●
●
● ●
● ●
●● ●
●
● ●
●
●
● ●
●● ●
● ● ●
●
●● ●● ●●
● ●
●
● ●
● ●
● ●
● ● ●● ●
● ● ●
● ●
●
● ●● ●
● ● ●
● ●
●
●
● ●
●
● ●
●
● ●
● ● ● ●
●
●
●●●
●●
● ● ●● ● ●
●● ●
● ●
● ● ● ●●
● ●
●● ● ●●
●
●●● ●● ●
● ● ●
●
● ● ●●
● ●● ● ●
● ●
●●
●● ●● ●
● ●
●
● ●
● ●
● ●● ●
●
● ●● ●
●
● ●● ● ● ●
● ● ●
● ●●
●
● ●●
●● ●●
● ●
●
● ● ●●
●
● ●● ● ● ● ●
● ●●
●
● ●● ●
●● ●
● ● ●
● ● ●●
● ●
● ● ●● ●
●● ●
● ●
● ● ●
● ●● ●●
●● ● ●
●●
● ●
time
10 ● ●●●
●● ● ● ●● ●
●●
● ● ●
●● ● ● ●●
● ●
● ●
●●● ●
●
● ●
●● ●
● ●
●
●
●
●● ●
●
● ● ●
●
●●
●● ● ●
●● ● ●
● ●
●●
●● ●●
●●
● ● ● ●
●● ● ●●
● ●
●
●● ● ●●
●● ●
●
●● ●
● ●● ●
● ●● ● ●●
●●
●
●
●● ●●
● ● ●
●
● ● ● ● ●●●
●
●●
●
●● ●
●
● ●
● ● ●
●
● ●
●● ●
● ●●
● ● ● ●●
●
●
●●
● ● ●
● ● ●
● ● ●
● ●● ●
● ● ● ●
● ● ●
● ●● ● ●
● ●
●
● ●
● ● ●
● ●●
●
● ●
●● ● ●
●
● ● ● ●
●
●
●
● ●●
●
5 ●
● ●
● ●
●
●● ●
● ●
● ●
● ●
● ●
● ●
●
●
● ● ●
●
●
●
0
100 200 300 400 500
day
Thursday, 19 March 2009

24.
70
60
50
40
count
30
20
10
0
0 5 10 15 20
time
Thursday, 19 March 2009

25.
20
●
●
●
●
15 ●
●
●
● ●
●
● ●
●
●
●
●
●
●
● ●
●
● ●
●
●
time
10 ●
● ●
● ●
●
●
●
●
●
●
●
●
●
●
5 ●
●
●
0
1.0 1.5 2.0 2.5 3.0 3.5 4.0
week
Thursday, 19 March 2009