Joint Probability Function

Study guide: Exam P-1

The joint probability function (joint PMF) of discrete random variables $X$ and $Y$ fully characterizes the Multivariate Distribution of $(X, Y)$ .

$p(x,y) = P(X = x,\; Y = y)$

It satisfies $p(x,y) \geq 0$ for all $(x,y)$ and $\displaystyle\sum_x \sum_y p(x,y) = 1$
Marginal Probability Functions are obtained by summing out one variable
Conditional Probability Functions are obtained by fixing one variable and dividing by its marginal, so the slice sums to 1: $p_{X \mid Y}(x \mid y) = \dfrac{p(x,y)}{p_Y(y)}$ for $p_Y(y) > 0$

Number of Claims and Policies Lapsed Let $X$ = number of claims (0 or 1) and $Y$ = policies lapsed (0 or 1). Joint PMF:

$Y=0$ $Y=1$

$X=0$ 0.50 0.20

$X=1$ 0.20 0.10

Answer $P(X=1, Y=1) = 0.10$ . Marginal $P(X=1) = 0.20 + 0.10 = 0.30$ .

Conditional: fixing $Y = 1$ leaves the slice $(0.20, 0.10)$ , which sums to $p_Y(1) = 0.30$ , so $P(X=1 \mid Y=1) = 0.10/0.30 = 1/3$ and $P(X=0 \mid Y=1) = 2/3$ .