Identifying random variables from moment generating functions and using characteristic functions Suppose that $X$ and $Y$ have moment generating functions $M_X(t),M_Y(t)$ respectively and $U$ has uniform distribution on $[0,1]$.
What is a random variable that is a function of X,Y and U such that it's moment generating function is 1) $\int_0^1 M_X(tu) \, du$ and 2) $\frac{M_X(t)+M_Y(t)}2$?
I know the linear properties of moment generating functions, such as $M_U(t) = \int_0^1 e^{ut}f(u)\,du$ and the properties $M_{X+Y}(t) = M_X(t)M_Y(t)$ as well as $M_{aX+b}(t) = e^{bt} M_X(at)$ but don't know how other properties would apply. Would the product of UV yield a moment generating function corresponding to $\int_0^1 M_X(tu) \, du$.
Any help would be really appreciated!

Suppose $W=\begin{cases} 1 & \text{with probability } 1/2, \\ 0 & \text{with probability } 1/2, \end{cases}$
$\vphantom{\dfrac11}$and $Z=WX + (1-W)Y,$ so that $Z=X$ if  $W=1$ and $Z=Y$ if $W=0.$ Then
$$
M_Z(t) = \operatorname{E}(e^{tZ}) = \operatorname{E}(\operatorname{E}( e^{tZ}\mid W)). 
$$
Oberserve that
$$
M_{Z\,\mid\, W=1}(t) = \operatorname{E}(e^{tZ}\mid W=1) = \operatorname{E}(e^{tX}) \\  M_{X\,\mid\,W=0}(t) = \operatorname{E}(e^{tZ}\mid W=0) = \operatorname{E}(e^{tY})
$$
so
$$
\operatorname{E}(e^{tZ}\mid W) = \begin{cases} \operatorname{E}(e^{tX}) & \text{with probability } 1/2, \\ \operatorname{E}(e^{tY}) & \text{with probability } 1/2. \end{cases}
$$
Therefore
$$
\operatorname{E}(\operatorname{E}(e^{tZ}\mid W)) = \frac 1 2 \operatorname{E}(e^{tX}) + \frac 1 2 \operatorname{E}(e^{tY}) = \frac{M_X(t) + M_Y(t)} 2,
$$
so
$$
M_Z(t) = \frac{M_X(t) + M_Y(t)} 2.
$$
The question involving $\displaystyle \int_0^1 M_X(tu)\,du$ is a bit subtler.
Note that
$$
M_X(tu)=\operatorname{E}(e^{tXu}) = \operatorname{E}(e^{tXU} \mid U=u),
$$
so
$$
M_X(tU) = \operatorname{E}(e^{tXU} \mid U).
$$
It follows that
\begin{align}
\int_0^1 M_X(tu) \,du = \operatorname{E}(M_X(tU)) = \operatorname{E}(\operatorname{E}(e^{tXU} \mid U)) = \operatorname{E}(e^{tXU}) = M_{XU}(t).
\end{align}
