CAS Exam 1Probability (P)

The joint density of $X$ and $Y$ is $f(x,y)=2$ for $0<x<y<1$ and zero otherwise. Calculate $E[X]$.

The joint density of $X$ and $Y$ is $f(x,y)=x+y$ for $0<x<1,\,0<y<1$. Calculate $\operatorname{Cov}(X,Y)$.

Let $X$ and $Y$ be independent Exponential($\theta$) random variables (same $\theta$). Let $R=X/(X+Y)$. Derive the distribution of $R$ and interpret the result in terms of how loss dollars split between two independent, identically distributed claimants sharing a fixed total.