09 unif exp gamma
TRANSCRIPT
Hadley Wickham
Stat310Uniform, Exponential, Gamma
Thursday, 5 February 2009
1. Exam note
2. Recap
3. Uniform distribution
4. Percentiles/quantiles
5. Exponential distribution
6. Gamma distribution
Thursday, 5 February 2009
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
exam01
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ABC
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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
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?
Thursday, 5 February 2009
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
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
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