joint probability density function?

Question

if X,Y and Z are RV's...and are related by the eqn Y=X+Z, how do you go about finding the joint pdf of x and y?
(by the way..X,Y and Z have an exponential distribution..but i dont even mind a general soln)

Answer 1

The general idea of getting a marginal is to integrate out the other dependence.

Integrate pdf(x,y,z) for z = 0 to z = y-x. You should end up with a function of x,y representing the joint pdf.

Answer 2

the issue does not specify a unfold for x; If the joint density f(x,y) = cy, If we assume (-? , ? ), the quintessential diverges ?? f(x,y) dxdy = one million is a call for. I even have assumed Y has achievable density f(y) = cy for 0 ? y ? 2 a) c? y dy = one million, 0 ? y ? 2 cy^2/2 = one million c/2 [ 2^2-0^2] = one million 2c=one million c=one million/2 f(y) = y/2 , 0 ? y ? 2 b) F(y) = P( Y ? y) = quintessential from 0 to y of y/2 quintessential of y/2 = y^2/4 Substituting the better cut back y and decrease cut back 0, F(y) = y^2/4 c) P(one million ? Y ? 2) = F(2)-F(one million) = 2^2/4 - one million^2/4 = one million - one million/4 = 3/4