Express the generating function of $Y=aX+b$ as a function of $M_x(t)$

Question

I need to express the generating function of $Y=aX+b$ as a function of $M_x(t)$.

For me, the generating function of a variable $Y$ is the expected value of $e^{ty}$, like this:

$$M_y(t) = E[e^{ty}] = E[e^{t(ax+b)}] = E[e^{tax}e^{tb}]$$

how to express it in function of $M_x(t)$? I thing I should write it as something related to $E[e^{tx}]$ but I don't know how to do it. Should I use some expected value property?

Answer 1

We have that

$$M_y(t) = \mathbb{E}_y[e^{ty}] = \mathbb{E}_x[e^{t(ax+b)}]$$ Note the variable that the expectation is taken over. We can simplify this to: $$M_y(t) = \mathbb{E}_x[e^{tax}e^{tb}] = e^{tb}\mathbb{E}_x[e^{(at)x}] = e^{tb}M_x(at)$$

We can take the "constant" out due to linearity of $\mathbb{E}$

Answer 2

$$M_Y(t) = E[ e^{tY}] = E[e^{t(aX+b)}] = E[e^{atX} e^{bt}] = e^{tb} E[e^{at X}] = e^{bt} M_X(at)$$