If X and Y are two indepnedent random variables, then

If X and Y are two independent random variables, then the probability distribution of the sum of X and Y is the convolution of their individual probability distributions. This is due to the fact that the outcome of one variable does not affect the outcome of the other variable.

In mathematical terms, if X and Y are independent, then the probability density function of (X+Y) is given by:

f(x+y) = ∫ fX(x) * fY(y) dx dy

where fX(x) and fY(y) are the probability density functions of X and Y respectively.

Also, the expected value of the sum of X and Y is the sum of their expected values. That is, E(X+Y) = E(X) + E(Y).

Similarly, the variance of the sum of X and Y is the sum of their variances. That is, Var(X+Y) = Var(X) + Var(Y). This is true because X and Y are independent, so there is no covariance term.