Is there any easy way to calculate the probability of the sum of two binomial random variable if the success rates of them are different each other?
I mean that $X \sim Bin(n,p_0)%$, $Y \sim Bin(m, p_1)$, $Z = X+Y$, $p_0 \neq p_1$ and hope to calculate the distribution function of $Z$.
I know the distribution of sum of random variables is calculated by convolution but I wonder if there is more easy way to get this. For example, $Z \sim Bin(m+n, p)$ if $p = p_0 = p_1$. Is there any similar formula in the case of $p_0 \neq p_1$?