Question

Let X and Y be a random variable with joint PDF:

f X Y ( x , y ) = { a y x 2 , x ≥ 1 , 0 ≤ y ≤ 1 0 otherwise

1. What is a?
2. What is the conditional PDF of given ?
3. What is the conditional expectation of given ?
4. What is the expected value of ?

Let X and Y be a random variable with joint PDF: fxv (, y) = {&, « > 1,0 Sysi 0 otherwise 1. What is a? 2. What is the conditional PDF fylx (x\y) of y given X = x? 3. What is the conditional expectation of Y given X? 4. What is the expected value of Y?

Answer 1

Given X & Y be rv. with joint pof f(214) osys ay x2 HTA نیوز By and condition of pdf; f fxy Cxivdxdy T ITS ay dedy #2 T ay [X2+] dy=1

MERCA PAGE NO CARE: Helyst af ydy [v 0 Facz की . We know that, for (aly) fxu (014) fic) fica sf (4) dy 12V dy - 2 ydy e2 . --56) 2. 471 . Conditional pdf of y given X = R is, fix=) 24x2 4* I left fux (Y/H) = 24 Osysi 3 conditional expectation of y_given X is; ELY 12= f'xi furox CM\x] dy ' y. 2y dy syte dy

- 3 Ely) 2 4 marginal density of y is given by fa(4) = 2 y de 22 = 24 -1 st f2 (4) = 24 - Expected value of Y is given by EUU) = 5* 2y dy 2 [x3] E(4) = 21 Pē. If xf y are independent Evi's, then, fly/X=2) = f(Y) is varified.