Moment generating function properties: $3φ_X (t)$ and $φ_X (t) × φ_X (6t)$

Question

Suppose that $φ_X (t)$ is the moment generating function of some random variable $X$. Are the following functions moment generating functions of some (other) random variables?

i. $3φ_X (t)$

I think this one isn't, because $3φ_X (t)=3E(e^{tX})=E(3e^{tX}), $ this can't be expressed as the expectation of an exponential power.

ii. $φ_X (t) × φ_X (6t)$

$φ_X (t) × φ_X (6t)=E(e^{tX})E(e^{6tX})=E(e^{tX+6tX})=E(e^{7tX})=φ_{7X} (t).$ I'm not too sure on whether I can merge the product of expectations into one expectation.

Is my reasoning for both the above statements correct?

Answer 1

Hint on i): if $\phi(t)$ is a moment generating function then what can be said about $\phi(0)$?

Hints on ii)

• What are the looks of moment generating function of $cX$ expressed in $\phi_X$ where $c$ is constant?
• If $X,Y$ are independent then what are the looks of the moment generating function of $X+Y$ expressed in $\phi_X$ and $\phi_Y$?