Bivariate Transformations

Theorem 4.3.1 If $X\sim Poisson(\theta)$ and $Y\sim Poisson(\lambda)$ and $X$ and $Y$ are independent, then $X+Y \sim Poisson(\theta + \lambda).$

Theorem 4.3.2 Let $X$ and $Y$ be independent random variables. Let $g(x)$ be a function only of $x$ and $h(y)$ be a function only of $y.$ Then the random variables $U=g(X)$ and $V=h(Y)$ are independent.