I would like to ask for help concerning this question lifted from the book An Introduction to the Theory of Statistics by Mood, Graybill, and Boes (2nd ed.).
Let $X_1$ and $X_2$ be independent standard normal random variables. Let $U$ be independent of $X_1$ and $X_2$, and assume that $U$ is uniformly distributed over $(0,1)$. Define $Z=UX_1+(1-U)X_2$. Find the distribution of $Z$.
Now, from what I understand, the conditional distribution of $Z$ given $U=u$ is normal with mean $0$ and variance $u^2+(1-u)^2$, so that:
$$f_Z(z)=\int_0^1\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{u^2+(1-u)^2}}e^{-\frac{z^2}{2[u^2+(1-u)^2]}}du$$
I would like to know if this correct or not? If yes, can anyone suggest how can I integrate this? Thanks.