I have a question:
$X$ is an exponential random variable with mean = 5. Let $Y=2X+3$. I'm required to find $M_Y(t)$, the moment generating function of $Y$.
Well I used the property that $M_Y(t)=M_{2X+3}(t)=M_X(2t)\cdot M_3(t)=e^{3t}\cdot M_X(2t)$ However, the difficulty I'm having is with computing $M_X(t)$. I am not sure whether to use $M_X(t)= \frac{1}{1-5t}~~~ \text{or}~~~ M_X(t)=\frac{5}{5-t}.$
Please, which one ought to be used in this case?
Thank you.