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- 1Chapter 9 - The Gamma Function The gamma function which is denoted Γ is given by? What are the four important properties of the gamma function?Γ(n) = integral [from 0 to infinity] x^(n-1)e^(-x) dx
1. Γ(n) = (n-1)Γ(n-1)
2. Γ(1) = 1
3. Γ(n) = (n-1)! for n=positive integer
4. Γ(1/2) = sqrt(n) - 2Chapter 9 - The Gamma Distribution A gamma random variable X~Ga(n,lambda) has PDF? State 2 results of the Gamma distributionfx(x) = [(lambda^n)/Γ(n)]*x^(n-1)e^(-lambdax)
1. alpha determines the shape and lambda affects the scale only
2. Ga(1,lambda) = Exp(lambda) - 3Chapter 9 - Expectation and Variance of Gamma Random Variables If X~Ga(n,lambda) then: E(X)=? Var(X)=? State the proof for the above statementsE(X) = n/lambda
Var(X) = n/lambda^2
Answer on separate sheet - 4Chapter 9 - MGF of Gamma Distribution The MGF of W~Ga(n,lambda) is? State the proof for the above statementMw(t) = [lambda/lambda-t]^n
Answer on separate sheet - 5Chapter 9 - MGF of Sum of Independent Exponential Distributions Let X1, X2, ..., Xn be iid Exp(lambda) random variables. Then Y= SIGMAXi~Ga(n,lambda). State this proofAnswer on separate sheet
- 6Chapter 9 - Beta Random Variables The Beta function is defined as? B(a,b)=?B(a,b) = integral [from 0 to 1] x^(a-1)*(1-x)^(b-1) dx
B(a,b) = [Γ(a)Γ(b)]/[Γ(a+b)] = [(a-1)!(b-1)!]/(a+b-1)! - 7Chapter 9 - The Beta Distribution A beta random variable X~Beta(a,b) has PDF? State 2 comments based on the beta distributionfx(x) = 1/(B(a,b))*x^(a-1)*(1-x)^(b-1) = [Γ(a+b)]/[Γ(aΓ(b)]*x^(a-1)*(1-x)^(b-1)
1. Beta(1,1) = U(0,1) distribution
2. If alpha = beta then the beta distribution is symmetric about x=1/2 otherwise the beta PDF is skewed - 8Chapter 9 - Expectation and Variance of Beta Random Variables If X~Beta(a,b) then: E(X)=? Var(X)=? State the proof of the above statementsE(X) = a/(a+b)
Var(X) = ab/[(a+b)^2(a+b+1)]
Answer on separate sheet - 9Chapter 9 - MGF of Beta Random Variable The MGF of a Beta random variable X~Beta(a,b) is given by?Mx(t) = 1 + SIGMAk=1[Π(a+r)/(a+b+r)]*t^k/k!
where Π has k-1 on top and r=0 on bottom
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- FlashCards sobre Sem título, criado por Cara Wales em 27-01-2016.
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- Cara Wales · jan 2016
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- 19 fev 2016
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