The question is:

Let X be a binomial random variable based on n trials, and a success probability of $\displaystyle p_x$; let Y be an independent binomial random variable based on m trials and a success probability of $\displaystyle p_y$. Find E(W) and Var(W), where W=4X+6Y.

(E(W) - expected value of W)

So from what I understand, that means $\displaystyle f_X(x)= \binom{n}{x}p^x(1-p)^{n-x}$ and the probability function for Y is similar.

I also know that given a variable say Z = X + Y, that $\displaystyle f_Z(z)=\int\limits_{-\infty}^{\infty} f_X(x)f_Y(z-x) \, dx$.

However since I have W=4X+6Y, I thought maybe then it should be something like $\displaystyle f_W(w)=\int\limits_{1}^{w} f_X(x)f_Y(\frac{w-4x}{6}) \, dx$. However that becomes a really ... nasty expression which I'm not sure how to simplify. I feel like I am doing something wrong.

I should be able to figure out the variance if I can do the expected value, because I should be able to just use the equation for variance.