Question

4. (14 pts) The joint pdf of X and Y is given by: (x + cya, 0 SX S1,0 Sys1 fxy(x, y) = otherwise For this question, it may be useful draw the region in the X, Y plane where the pdf is non- zero to help you determine the limits of the integrals. (a) Find the value of the constant c. (b) Find the marginal pdfs of X and Y, respectively. (c) Find the probability that both X and Y are less than (d) Find the conditional pdf of X given Y, fxy(x|y). Plot the pdf for y = (e) Find the conditional expectation of X given Y, E[XİY]. Plot E[X|Y] for Y from 0 to 1.

Answer 1

x + y2 Oct 31 oz ye! otherwise For this question , it may be useful draw the in the x,y plane where the pdf is ron-3000 to you determine the limits of the integrals. region help 1. f(x,y) = x + cy² aaxal ocykl Solo x + cya dx.dy = 1 do Czy + Cys de Cola C f(x) = ( x + 2 y z dy U 3 ys I = x+1/ 0<xkl. a 3 JO

Ply) - 42+aya de OLYKI P ( x < 7 , y 2 ) - "* + y de dy - " [24 +22) TOY xt CY 8x3 1 - 2 = 0.093

x+ 3/2 y ² 1/2 + 3y² .: OLX - 11B x + a 1/2+1/2 f (*/12) = x + 1 / :.olx. 21 Fle F(xly) = x+3y2 1/2+3/292 - x + 1/2 -1'**doocy24 de OLYLI . lix (*+/2) dx 0.588.