Let X and Y be independent uniformly distributed random vari

Let X and Y be independent uniformly distributed random variables on (0,1). Find the joint pdf of W = X + Y andV = X-Y.

Solution

Given that x and Y are independent and uniform on (0,1)

f(x) = 1, 0<x<1

f(y) = 1 , 0<y<1

Since x and y are independent,

joint pdf = f(x,y) = 1,

0<x<1 , 0<y<1

--------------------------------------------------------

W = X+Y is uniform in the interval (0,2) and

V = X-y is uniform in the interval (-1,1)

Pdf of W = 1/2, 0<w<2

and Pdf of V = 1/2, -1<v<1

Since x and y are independent, so also X+y and x-y

Hence joint pdf of W and V

= 1/4, 0<w<2, -1<v<1