Question

Let X and Y have the joint pdf f(x,y) = e^-x-y I(x > 0,y > 0).

a. What are the marginal pdfs of X and Y ? Are X and Y independent? Why?

b. Please calculate the cumulative distribution functions for X and Y, that is, find F(x) and F(y).

c. Let Z = max(X,Y), please compute P(Z ≤ a) = P(max(X,Y) ≤ a) for a > 0. Then compute the pdf of Z.

Answer 1

f(x,y) = exy , x>0, yo (a) Marginal pdf of X £.(a) = (f(ds = j *% d an feidy =ē ( (0+1) = ēx So, fx(x) = ēt x>0. get marginal with exactly similar calculation, we pdf of 7 as, f(3) = et y>o. Note that f(x). Fy (Y) = ex et - et y = f(x,y). So, X and y are independent. COF (6) cor of xi F(x) = I felt) dt F(x) = f et dt = - [et] 2 = 1-ex Similarly, cor of y is F(Y) = 1- et

and Y are independent} let z= max(x, y) P(z = a) = (max(x,y) ca). = P(xca, ya) = P(x sa) Plysa) (since, X = (a). F, (a) = (1-2 a) (1-8a) = (1- ea)? So, Ę(a) = (-ea)? azo. pdf of 2 is f(a) = f(a) = (-ėg)? da da - - 21- ea). Za 2 ē9 (1-) azo.