Given $X$ is a random variable with moment generating function $M(t)$, I am suppose to determine whether $M(t)M(5t)$ is the MGF of some random variable. The solution in my textbook says it is, if I let $Y=5X$ then $M(5t)$ is the MGF for $Y$ and thus $M(t)M(5t)$ is the MGF for $X+Y$, since moment generating functions are multiplicative.
But I looked up the proof that the MGF of $X+Y$ is the product of the MGF of $X$ and the MGF of $Y$, and it seems to follow from the independence of $X$ and $Y$. Here it is:
$$M_{X+Y}(t) = E[e^{(X+Y)t}] = E[e^{tX}e^{tY}].$$ Recall that $$E[g_1(X)g_2(Y)] = E[g_1(X)]E[g_2(Y)] \therefore E[e^{tX}e^{tY}] = E[e^{tX}]E[e^{tY}] = M_X(t)M_Y(t)$$
The step after "Recall that" relies on indendence of $X$ and $Y$. So in my problem, where $Y=5X$, this proof does not hold and thus I need another reason to believe that I can multiply these MGFs together. Thanks!
Best Answer
You are right that the argument given is incorrect since $X + Y = 6X$ and so unless $M(5t)M(t) = M(6t)$ for every $t$ we will not have that the mgf of $X+Y$ is $M(t)M(5t)$.
However, it is always possible (and I think what is intended in the argument you mentioned) to take some probability space with random variables $\tilde{X}$ and $Y$ defined on it such that $\tilde{X}$ and $Y$ are independent and such that $X \stackrel{d}{=} \tilde{X}$ and $Y \stackrel{d}{=} 5X$. This then gives the result since the mgf of a random variable is determined by its distribution and, as you noted, the mgf of a sum of independent random variables is the product of their mgfs.