Let X and Y be two independent random variables, X ∼ Γ(α, λ) and Y ∼ Γ(β, λ).Find the joint probability density function f(Z,W)of the vector (Z, W)(b) Show that Z and W are independent(c) Show that Z ∼ Γ(α + β, λ) and W ∼ B(α, β) 5. Let X and Y be two independent random variables, X ~Г(а, 1) and Y ~ Г(B, ).fx(x) = Fde" (Aæ)a-1, x >0{Ta le-dzotherwiseLet Z = X +Y and W =X+Y•(a) Find the joint probability density function f(z.w)of the vector (Z, W)(b) Show that Z and W are independent(c) Show that Z ~ r(a+ B, X) and W ~ B(a, ßB)

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Answered: Let X and Y be two independent random…

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Let X and Y be two independent random variables, X ∼ Γ(α, λ) and Y ∼ Γ(β, λ).

Find the joint probability density function f(Z,W)of the vector (Z, W)
(b) Show that Z and W are independent
(c) Show that Z ∼ Γ(α + β, λ) and W ∼ B(α, β)

5. Let X and Y be two independent random variables, X ~
Г(а, 1) and Y ~ Г(B, ).
fx(x) = Fde" (Aæ)a-1, x >0
{
Ta le-dz
otherwise
Let Z = X +Y and W =
X+Y•
(a) Find the joint probability density function f(z.w)of the vector (Z, W)
(b) Show that Z and W are independent
(c) Show that Z ~ r(a+ B, X) and W ~ B(a, ßB)

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