Are $ 5 \text{ x } m_y(s)$ and $m_y(s)^2$ moment generating functions

Question

Say if I had some MGF like $m_y(s)$ of some random variable $Y$, are $ 5 \text{ x } m_y(s)$ and $m_y(s)^2$ moment generating functions? This is a curiosity thing, does multiplying through by some integer have an effect? How about squaring the MGF?

Answer 1

In general no, a scalar multiple of a Moment Generating Function (MGF) or a square of an MGF will not be a MGF. Consider the MGF of a standrd normal density

$M_X(t)=e^{t^2/2}.$

Then the scalar multiple in the post would be

$5e^{t^2/2}$ which is not an MGF.

If you multiply your random variable by a constant then the resulting MGF of that random variable $Y=5X$ (say) would be

$\int e^{t5x}f(x)dx = \int e^{t^\prime x}f(x)dx = M_X(5t) \neq 5M_X(t)$ unless of course your MGF was a linear function in $t$.