Let $U\sim \mathcal U(0, 1)$ be a random variable uniformly distributed over the interval $[0, 1]$. Let $X_1, X_2\sim \Gamma(a, b)$ be two iid random variables with a Gamma distribution. Now it is straightforward to calculate the mean and variance of the random linear combination $$Y = U\cdot X_1 + (1-U)\cdot X_2,$$ which is simply $\mathbb E(Y)=\mathbb E(X_1)$ and $var(Y)=\frac23 var(X_1)$. However, it would be very interesting for me to know the precise distribution of $Y$ but I didn't find any result for this particular problem. Does anyone know what the distribution is of $Y$?