Sum and Product of Discrete Random Variables

Theorem

Let $X$ and $Y$ be discrete random variables on the probability space $\left({\Omega, \Sigma, \Pr}\right)$.

Sum of Discrete Random Variables

Let $U: \Omega \to \R$ be defined as:

$\forall \omega \in \Omega: \map U \omega = \map X \omega + \map Y \omega$

Then $U$ is also a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.

Product of Discrete Random Variables

Let $V: \Omega \to \R$ be defined as:

$\forall \omega \in \Omega: \map V \omega = \map X \omega \map Y \omega$

Then $V$ is also a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.