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Hadley Wickham Stat310 Uniform, Exponential, Gamma Thursday, 5 February 2009
19

09 Unif Exp Gamma

May 25, 2015

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Page 1: 09 Unif Exp Gamma

Hadley Wickham

Stat310Uniform, Exponential, Gamma

Thursday, 5 February 2009

Page 2: 09 Unif Exp Gamma

1. Exam note

2. Recap

3. Uniform distribution

4. Percentiles/quantiles

5. Exponential distribution

6. Gamma distribution

Thursday, 5 February 2009

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Graded and ready for you to pick up when class is over. Model answers on website (will be improved asap)

See me in office hours if any questions.

Technique: From general to specific.

When are you finished?

Exam notes

Thursday, 5 February 2009

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exam01

count

0

5

10

15

0 10 20 30 40 50

ABC

Thursday, 5 February 2009

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If X is a continuous rv, what is f(x)? What is F(x)

Give two ways to compute P(a < X < b)

What are the definitions of expectation, mean, variance and moment generation function?

Recap

Thursday, 5 February 2009

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Recap

Let X be a discrete rv with pmf:

fX(1) = 0.25 fX(2) = 0.5 fX(3) = 0.25

Let Y be a continuous rv with pdf:

fY(y) = 1/2 x ∈ [1, 3]

Sketch a graph of fX and fY. Sketch a graph of FX and FY. What’s another reason Y is called continuous?

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X is discrete X is continuous

f(x) is pmf f(x) is pdf

P (X ! A) =!

x!A

f(x) P (X ! A) =!

x!Af(x)

sums integrals

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Example

f(x) = x e-x x > 0

Calculate F(X)

Is f a pmf?

Calculate mean

Calculate mgf

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Assigns probability uniformly in an interval [a, b] of the real line

X ~ Uniform(a, b)

The uniform

f(x) =1

b! ax " [a, b]

F (x) = 0 x < a

F (x) = x!ab!a x ! [a, b]

F (x) = 1 x > bThursday, 5 February 2009

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Intuition

X ~ Unif(1, b)

What do you expect the mean of X to be?

What about the variance?

How does this compare to the discrete case?

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E(X) =a + b

2

V ar(X) =(b! a)2

12

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Question

X ~ Unif(0, 1)

Y = 10 X

What is the distribution of Y?

How does the variance of Y compare to the variance of X?

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Percentiles

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Quantiles

F(x) lets us find probability easily.

What if we have probability and want to find x? i.e. given p we want to find x such that F(x) = p

Need to invert function (or use table)

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Special quantiles

Lower quartile = F-1(0.25)

Median = F-1(0.5)

Upper quartile = F-1(0.75)

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Exponential distribution

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Gamma

New distribution.

X ~ Poisson(λ). Let Y be the time you wait until α events occur occurs.

Then Y ~ Gamma(θ, α), θ = 1/λ

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f(x) =1!e!x/!

M(t) =1

1! !t

Exponential Gamma

pdf

mgf

Mean

Variance

!

!2

f(x) =1

!(!)"!x!!1e!x/"

M(t) =1

(1! !t)!

!"

!"2

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Homework

Thursday, 5 February 2009