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I am not looking for just free handout answers but I am having extreme trouble with quite a few Prob and Stat problems. My instructor likes to do proofs in class and then assign application problems, up until now I have been able to read the book to figure the work out but I cannot find these questions in the book as they are ones he posted on an online homework site instead.

The moment generating function of X is MX(t)=e^(2e^t − 2) and Y is MY(t)=(0.2e^t + 0.8)^7. Suppose X and Y are independent and compute the following: Pr(X+Y=3)= ?

I think this is a poisson random variable mixed with a binomial but I don't know how to solve it

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If you know that $X, Y$ are discrete random variables with support on non-negative integers only, then you may also try the probability generating function technique without reverting to the original pmf and do convolution.

Note that the probability generating function satisfy $G_X(z) = M_X(\ln z)$. So by independence, $$G_{X+Y}(z) = G_X(z)G_Y(z) = M_X(\ln z)M_Y(\ln z)$$ and thus

$$ \Pr\{X + Y = 3\} = \frac {G^{(3)}(0)} {3!}$$

Of course differentiating this product $3$ times also requires some effort (hopefully not too tedious). Should be similar to the effort when you recognize the original pmf quickly and do convolution.

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