17 normal intro

Hadley Wickham

Stat310The normal distribution

Thursday, 12 March 2009

1. Exam

2. Recap

3. Finish off convergence example

4. The normal distribution (reading?)


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”

Exam


Question 3


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0.2 0.4 0.6 0.8 1.0


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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?

Recap


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?


limn!"

P (|Zn ! p| " !) = 1

!! > 0


Time Event Total Estimate

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


n

est

0.0

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000


1000 runs


n

est

0.0

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000

99%

1%

75%

25%


n

est

0.20

0.25

0.30

0.35

0.40

200 400 600 800 1000

99%

1%

75%

25%

50%


n

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99%

1%

75%

25%

50%


The normal distribution


f(x) =1!2!

e!(x!µ)2

2!2

Is this a valid pdf?


M(t) = eµt+ 12 !2t2

See book for derivation. A few tricks + lots of algebra


Your turn

If X ~ Normal(μ, σ2), what is the mean and variance of X?

(Work it out - don’t just write down what you know!)


f(x)

0.0

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−10 −5 0 5 10

f(x)

0.0

0.1

0.2

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−10 −5 0 5 10

f(x)

0.0

0.1

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−10 −5 0 5 10

f(x)

0.0

0.1

0.2

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−10 −5 0 5 10

N(-2, 1) N(5, 1)

N(0, 1)

N(0, 16)N(0, 4)

f(x)

0.0

0.1

0.2

0.3

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−10 −5 0 5 10


Transformations

If X ~ Normal(μ, σ2), and Y = a(X + b)

Y ~ Normal(b + μ, a2σ2)

If a = -μ and b = 1/σ, we often write

Z = (X - μ) / σZ ~ Normal(0, 1) = standard normal


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)


P (Z < z) = !(z)

P (!1 < Z < 1) = 0.68P (!2 < Z < 2) = 0.95P (!3 < Z < 3) = 0.998

!(!z) = 1! !(z)


Readings: 5.3, 5.4, 5.5


17 normal intro

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