Question

Suppose the continuous random variables X and Y have joint pdf: fXY (x, y) = （1/2）xy for 0 < x < 2 and x < y < 2 (a) Find P(Y < 2X) by integrating in the x direction first. Be careful setting up your limits of integration. (b) Find P(Y < 2X) by integrating in the y direction first. Be extra careful setting up your limits of integration. (c) Find the conditional pdf of X given Y = y, fX|Y =y(x). Be sure to state the support (and the values y that can be conditioned on). (d) Find the expected value of X given Y = y, E(X|Y = y). (Be sure to state the values y that can be conditioned on.) (e) Using your answer from (1h), find the conditional pdf of X given Y = 1, fX|Y =1(x). Be sure to state the support. (k) Using your answer from (1i), find the expected value of X given Y = 1, E(X|Y = 1). (l) Find P(X < Y < √ 2X) by integrating in the x direction first. Be careful setting up your limits of integration. (m) Find E(X/Y ). (n) Find the marginal pdf of X, fX(x). Be sure to state the support. (o) Are X and Y independent? Why?

Answer 1

i first did part 'b' and then part 'a' and some parts are very trivial like the last part 'o' obviously X and Y are dependent as in joint pdf range of X is dependent on Y

the following 2 images have the solutions check them out...We are allowed to do first 4 parts if a question has more than 4 parts but i still did few extra too

but if you have problem with others just drop a comment

image 1:

image 2:

cheers :)