Let Y be exponentially distributed with parameter 1 and let

Let Y be exponentially distributed with parameter 1, and let Z be uniformly distributed over the interval [0,1]. Assume Y and Z are independent. Find the probability density functions of W = Y - Z and X = |Y - Z|.

Solution

W = Y-Z

and X = |Y-Z|

As Y and Z are independent

pdf of Y = e^-Y, Y>0

pdf of Z = 1

As e^-y is always positive with e and y positive

we can find that e^-y <1 for all y

Hence pdf of W = e^-W , W >0

and PDF of X = |Y-z|

Thus X =x for Y>Z and x = -x, for Y<z