Conditioning and Independence for Discrete Random Variables

"Given that this name has already downgraded once, what's the distribution of further downgrades?" is a conditional-PMF question. "Are order-arrival counts on two different venues independent?" is an independence-of-random-variables question. This note completes the discrete-random-variable toolkit by restating Probability Foundations's conditioning and independence machinery in random-variable notation: the conditional PMF given an event or given another random variable, the sequential construction of a joint PMF via conditioning (the multiplication rule, restated), the total probability formula for PMFs (a divide-and-conquer route to a marginal), and independence of random variables — including the algebraically useful fact that independence makes $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$, in sharp contrast to Joint PMFs's linearity-of-expectation result, which needed no such assumption.

Learning objectives. By the end of this note you should be able to: