Question

4.4-2. Let X and Y have the joint pdf f(x, y) r + y, = x + y, (a) Find the marginal pdfs fx(t) and fy (v) and show that f(x,y)关fr (x)fy(y). Thus, X and Y are dependent. (b) Compute (i) μ x, (ii) μ Y. (111) 07, and (iv) 어.

Answer 1

f(x,y) = x + y ; 0 ≤ x ≤ 1 ; 0 ≤ y ≤1

(a) marginal pdf of x is

f(x) = $\int_{0}^{1} f(x,y) dy$

f(x) = $\int_{0}^{1} (x + y) dy$

f(x) = [xy + y²/2]¹₀

f(x) = x + 1/2 ; 0 ≤ x ≤ 1

similarly,

f(y) = y + 1/2 ; 0 ≤ y ≤ 1

now we can easily see that

f(x,y) ≠ f(x) f(y) so X and Y are dependent her e

(b) (i) E(X) = $\int_{0}^{1} xf(x) dx$ = $\int_{0}^{1} x(x + 1/2)dx$ = $\int_{0}^{1} (x^2 + x/2)dx$

E(x) = [x³/3 + x²/4]¹₀ = 1/3 + 1/4 = 7/12

similaryl ,

(ii) E(Y) = 7/12

(iii) Here Var(X) = E(X²) - E(X)²

E(X²) = $\int_{0}^{1} x^2f(x)dx$ = $\int_{0}^{1} x^2 (x + 1/2)dx$ = $\int_{0}^{1} (x^3 + x^2/2)dx$

E(X²) = [x⁴/4 + x³/6]¹₀ = 1/4 + 1/6 = 10/24 = 5/12

Var(X) = 5/12 - (7/12)² = 11/144

(iv) Var(Y) = 11/144