Consider three independent identically distributed real random variables $X,Y,$ and $Z$. Assume that $X= U(Y+Z)$ where $U$ is a uniform random variable on $[0,1]$. Find the distribution of $X$.
Here is my attempt: Let $f(x)$ denote the pdf of $X,Y,Z$.
$\begin{eqnarray*} P(X=x) = P(U(Y+Z)=x)&=& \int_0^1 P(Y+Z=x/u)P(U=u)du\\ &=& \int_0^1 P(Y+Z=x/u)du\\ &=& \int_0^1 \int_{-\infty}^\infty P(Y=y)P(Z=x/u-y)dydu\\ &=& \int_0^1 \int_{-\infty}^\infty f(y)f(x/u-y)dydu \end{eqnarray*}$
Therefore we have $f(x) =\int_0^1 \int_{-\infty}^\infty f(y)f(x/u-y)dydu $ and I'm not sure what to do from here.