[Math] The sum of moment generating functions

Question

Let $X, Y$ be independent r.v with moment generating functions $M_X(t)$ and $M_Y(t)$ respectively. Is there a function of $X$ and $Y, Z$, with moment generating function $$\frac{M_x(t) + M_y(t)}2$$

Answer 1

Let $Z$ be independent of $X$ and $Y$ such that $P(Z=1)=P(Z=0)=\tfrac12$ and put $U=XZ+Y(1-Z)$. Then $$ \begin{align} M_U(t)&=\mathrm{E}[\mathrm{e}^{tU}]=\mathrm{E}[\mathrm{e}^{tX}\mathbf{1}_{Z=1}]+\mathrm{E}[\mathrm{e}^{t Y}\mathbf{1}_{Z=0}]\\ &=\tfrac12M_X(t)+\tfrac12 M_Y(t). \end{align} $$