Joint PMFs of Multiple Random Variables

A portfolio's dollar return is not one random variable's business — it is a function of every position's return at once, $R_p = \sum_i w_i R_i$. Reasoning about several random variables simultaneously needs a joint description, and this note builds it: the joint PMF $p_{X,Y}(x,y)$, which captures everything about how $X$ and $Y$ behave together; the marginal PMFs recovered by summing the joint table along rows or columns; and the expected value rule for functions of several random variables, whose most useful special case is the linearity of expectation — $\mathbb{E}[aX+bY+c]=a\mathbb{E}[X]+b\mathbb{E}[Y]+c$ — which holds with no independence assumption whatsoever, making it the single most load-bearing identity for aggregating risk across positions.

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