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where Γ(ω):=∞vω−1e−vdv is the Γ(α)...

Let X ∼Γ(α,β), which has p.d.f. fX(x)= βα xα−1e−βxI(0,∞)(x), where Γ(ω):=∞vω−1e−vdv is the Γ(α) 0

Gamma function, for ω > 0.

1. Compute the moment generating function of X. [5 points] [Hint: 0∞ fX(x)dx=1]

2. Compute the first moment of X in two ways:

(a) Solve the integral 0∞xfX(x)dx. [5 points] [Hint: Γ(ω+1)=ωΓ(ω)]

(b) Use the moment generating function. [5 points]

3. Compute the variance of X by using the moment generating function. [5 points]

4. Let X1∼Γ(α1,β) and X2∼Γ(α2,β) be independent random variables. Find the p.d.f. of Y =X1+X2 in two ways:

(a) Use moment generating functions. [5 points] (b) Use the approach with Jacobian. [5 points]

K

5. Let X1,...,XK ∼ Γ(γ,δ). What is the distribution of Z = Xi? [5 points] [Hint: use part 4.]

Gamma function, for ω > 0.

1. Compute the moment generating function of X. [5 points] [Hint: 0∞ fX(x)dx=1]

2. Compute the first moment of X in two ways:

(a) Solve the integral 0∞xfX(x)dx. [5 points] [Hint: Γ(ω+1)=ωΓ(ω)]

(b) Use the moment generating function. [5 points]

3. Compute the variance of X by using the moment generating function. [5 points]

4. Let X1∼Γ(α1,β) and X2∼Γ(α2,β) be independent random variables. Find the p.d.f. of Y =X1+X2 in two ways:

(a) Use moment generating functions. [5 points] (b) Use the approach with Jacobian. [5 points]

K

5. Let X1,...,XK ∼ Γ(γ,δ). What is the distribution of Z = Xi? [5 points] [Hint: use part 4.]

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