# Convolution theorem - Sum of random variable and constant

Problem:

Random variables X, Y are independent. X has normal distribution with parameters (a= 1, b^2 = 4). Y has also normal distribution, but Y is equal to (-X). Find all constants c and k > 0,in order to find X + Y + c and k*X so they would have same distribution.

I found out, it has to be done throught convolution theorem. So first I will sum two random variables and then compute constant c. But I don't really know how to compute the constant. My idea is to use definite integral of f(z) + c = 1 (f(z) is X+Y). Is it correct?

• Sorry, but how can $X,Y$ be independent if $Y=-X$? Do you mean that it has the same distribution as $-X$? – Clement C. Dec 11 '16 at 19:53