Joint Probability Functions and Joint CDFs for Discrete Random Variables

Pricing a joint-life policy requires a joint probability model for two correlated lifetimes. Get the joint distribution wrong and the premium is off by tens of thousands. Exam P tests whether you can build, read, and extract probabilities from joint PMF tables.

For two discrete random variables $X$ and $Y$, the joint PMF is:

$$p_{X,Y}(x,y)=P(X=x,Y=y)$$

Two properties must hold:

• $p_{X,Y}(x,y)\ge 0$ for all $(x,y)$
• ${\sum }_x{\sum }_y{p}_{X,Y}(x,y)=1$

On the exam, joint PMFs appear as tables. Each cell holds the probability of a specific $(x,y)$ pair. Row totals and column totals give the marginal distributions.

HIGH-FREQUENCY: Given a table, computing $P(X+Y\le c)$, $P(X>Y)$, or $P(X\le a,Y\le b)$ by summing appropriate cells is a core skill.

The joint CDF of $X$ and $Y$ is:

Common mistakes

• Forgetting that $X+Y$ is bounded by the sample size. With 2 draws, $X+Y\le 2$. Assigning nonzero probability to cells like $(2,1)$ produces probabilities summing above 1.
• Confusing $F_{X,Y}(x,y)$ with $F_X(x)\cdot F_Y(y)$. The joint CDF equals the product of marginal CDFs only under independence. In Example 1: $F_X(1)\cdot F_Y(1)=(5/6)(5/6)=25/36$, but $F_{X,Y}(1,1)=2/3=24/36$. Trap: 25/36.
• Summing wrong cells for diagonal events. For $P(X+Y\le 2)$, including cell $(2,1)$ where $x+y=3$ inflates the answer. Verify each cell's coordinates before including it.

Bottom line

• Joint PMF: $p_{X,Y}(x,y)=P(X=x,Y=y)$, must be non-negative and sum to 1
• Joint CDF: $F_{X,Y}(x,y)=P(X\le x,Y\le y)$, cumulate down and to the left across every qualifying cell
• Rectangle formula: $P(a<X\le b,c<Y\le d)=F(b,d)-F(a,d)-F(b,c)+F(a,c)$
• Independence test: $p_{X,Y}(x,y)=p_X(x)\cdot p_Y(y)$ must hold for EVERY cell

Exam shortcut

When handed a joint PMF table, immediately compute row and column totals in the margins. For diagonal events like $P(X+Y\le k)$, draw a diagonal line across the table to separate qualifying cells visually. "CDF = Cumulate Down and Left, Four terms for rectangles." $F(b,d)-F(a,d)-F(b,c)+F(a,c)$. "Every cell must multiply out" for independence. Row total times column total. One mismatch proves dependence.

The full lesson (about 3,020 words, 20 min read) adds 3 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 3a

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