A communication channel has random channel variable X random

A communication channel has random channel variable X, random noise input N and output Y = X+N. Suppose X = 1 with probability 0.5, X = -1 with probability 0.5 and N has Laplacian density function:

f(n) = .5*alpha*e^(-alpha*|n|)

(b) Find the probability density function fY(y) for Y .

Solution

y = X+N

P(X=1) =0.5 and P(x=-1) = 0.5

f(n) = 0.5*alpha*e^(-alpha*|n|)

f(Y-x) = 0.5*alpha*e^(-alpha*|n|)

f(Y+1) = 0.5alphae^(-alpha*|Y+1|)

Or f(y) = 0.25alphae^(-alpha*|Y|