Conditional and Marginal Probability Functions for Discrete Random Variables

"Given that this policyholder filed 3 hospital claims, what is the distribution of their prescription claims?" That requires a conditional distribution, the joint PMF sliced to one row and rescaled.

Given the joint PMF $p_{X,Y}(x,y)$, the marginal PMF of $X$ sums over all values of $Y$:

$$p_X(x) = \sum_y p_{X,Y}(x,y)$$

In a table, the marginal of $X$ is the column of row totals. The marginal of $Y$ is the row of column totals. They sit in the margins, hence the name.

HIGH-FREQUENCY: Computing marginals by summing rows or columns is tested on virtually every joint distribution problem.

The conditional PMF of $Y$ given $X = x$:

$$p_{Y|X}(y|x) = \frac{p_{X,Y}(x,y)}{p_X(x)}, \quad p_X(x) > 0$$

This isolates one row of the table and rescales so the entries sum to 1. The rescaling factor is $1/p_X(x)$.

Symmetrically, $p_{X|Y}(x|y) = p_{X,Y}(x,y) / p_Y(y)$ divides by the column total.

Common mistakes

• Dividing by the wrong marginal. For $p_{Y|X}(y|x)$, divide by $p_X(x)$, not $p_Y(y)$. In Example 1: $p_{Y|X}(1|2) = (6/20)/(8/20) = 3/4$. Dividing by $p_Y(1) = 14/20$ instead gives 6/14 = 3/7. Trap: 3/7.
• Forgetting a term in the law of total probability. With three classes, omitting the high-risk term $(3/4)(1/6) = 1/8$ gives $P(Y=1) = 1/10 + 1/6 = 4/15 \approx 0.267$ instead of 47/120 = 0.392. Trap: 4/15.
• Confusing $P(X|Y)$ with $P(Y|X)$. These have different denominators. $P(Y=1|X=3) = 3/4$, but $P(X=3|Y=1) = 15/47$. Using one as the other is a Bayes error.

Bottom line

• Marginal PMF: sum the joint over the other variable (row totals give the marginal of X, column totals give the marginal of Y).
• Conditional PMF: $p_{Y|X}(y|x) = p_{X,Y}(x,y) / p_X(x)$, divide each cell by its row total.
• Joint = Conditional times Marginal: $p_{X,Y}(x,y) = p_{Y|X}(y|x) \cdot p_X(x)$.
• Law of total probability: $p_Y(y) = \sum_x p_{Y|X}(y|x) \cdot p_X(x)$.

Exam shortcut

When a problem gives conditional distributions and a marginal (the "Bayesian setup"), build the full joint PMF table first, multiply each conditional by its marginal weight. Then read off whatever the question asks. "Joint = Conditional times Marginal" (JCM). To get C: divide by M. To get M: sum across the other variable. "Divide by the given." $P(Y|X)$ divides by the row total. $P(X|Y)$ divides by the column total.

The full lesson (about 2,754 words, 18 min read) adds 3 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 3b

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