Moment generating function, change of variable

Question

If $X=aY$ where $X, Y$ are random variables and $a$ is a constant. Do I have $$M_X(t)=M_Y(t/a)$$ where $M$ is the moment generating function. How can I prove it?

Answer 1

Along the lines of what the commenter fgp said, we can go right to the definition of the moment generating function, $M_X(t) = E(e^{tX})$. Then we just use the fact that $X = aY$ as follows:

\begin{align*} M_X(t) &= E(e^{tX})\\[5pt] &= E(e^{t(aY)})\\[5pt] &= E(e^{(ta)Y}) \\[5pt] &= M_Y(ta) \end{align*}

Hope that helps.