Joint p.d.f. 2

(a)(i) f(x1) = 1 (0 <= x1 <= 1); f(x2) = 1/2 (0 <= x2 <= 2)

(ii) f(x1,x2) = 1/2

(iii) Let y1 = x1 + x2, y2 = x1

Then x1 = y2, x2 = y1 - y2

Jacobian = |J| = 1

So, f(y1,y2) = 1/2 * 1 = 1/2

f(y1) = ∫ 1/2 dy2

when 0<= y1 <= 1, 0 <= y2 <= y1

And then f(y1) = ∫ 1/2 dy2 = y1/2

If 1 <= y1 <= 2, 0 <= y2 <= 1 (Of course y2 <= y1 but we do not take the upper as y1)

So, f(y1) = ∫ 1/2 dy2 = 1/2

If 2 <= y1 <= 3, y1 - 2 <= y2 <= 1

So, f(y1) = ∫ 1/2 dy2 = (1/2)(3 - y1)

(b)(i) The moment generating funvtion of X is

E[exp(tx)]

= ∫ x^(k - 1)exp[(t - λ)x] λ^k/ Γ(k) dx

Let y = (λ - t)x, dy = (λ - t)dx

= 1/(λ - t) ∫ [y/(λ - t)]^(k - 1)exp(-y) λ^k/ Γ(k) dy

= λ^k/(λ - t)^k ∫ y^(k - 1)exp(-y) / Γ(k) dy

= [λ/(λ - t)]^k

Since the integral is equal to 1 by the definition of Γ(k)

(ii) E(X) = M'(0) = kλ^k/(λ - t)^(k + 1) = k/λ

E(X^2) = M''(0) = k(k + 1)/λ^2

Var(X) = E(X^2) - [E(X)]^2 = k/λ^2