Conditional Distributions

Core Idea

The conditional distribution of X given Y=y is the distribution of X when Y is fixed: P(X=x|Y=y) = P(X=x,Y=y)/P(Y=y). Conditional distributions capture how one variable's distribution depends on another's value.

How It's Best Learned

From a joint distribution table, select a column or row and normalize it to sum to 1. For continuous distributions, condition by dividing joint PDF by marginal PDF. Compare conditional distributions for different values.

Explainer

You know how to work with joint distributions — probability tables or density functions describing two random variables simultaneously — and you know conditional probability: P(A|B) = P(A∩B)/P(B). Conditional distributions combine these ideas. Instead of asking for the probability of a single event given another event, you ask: what does the entire distribution of X look like when Y is fixed at a specific value?

In the discrete case, the idea is concrete. Suppose (X, Y) has a joint probability table. Fix a particular value y for Y. The conditional distribution of X given Y = y is the distribution you get by looking only at the column (or row) of the table where Y = y, then rescaling so the values sum to 1. Formally: P(X = x | Y = y) = P(X = x, Y = y) / P(Y = y). The denominator P(Y = y) is the marginal probability of that value of Y, obtained by summing the column — exactly the marginal distribution you know how to compute. Dividing by this sum is the normalization step: you zoom in on the subpopulation where Y = y and rescale to form a valid probability distribution for X within that subpopulation.

In the continuous case, individual values have probability zero, so the formula P(X = x | Y = y) / P(Y = y) would be 0/0. Instead, the conditional density is defined as the ratio of densities: f_{X|Y}(x|y) = f_{X,Y}(x, y) / f_Y(y). This is the same logical structure — numerator is joint, denominator is marginal — just expressed in terms of density functions rather than probabilities. To verify it is a valid density: integrate over x, getting ∫ f_{X,Y}(x,y) dx / f_Y(y) = f_Y(y) / f_Y(y) = 1. The normalization works out automatically from the definition of the marginal.

Conditional distributions reveal the dependence structure between variables. If X and Y are independent, the conditional distribution of X given Y = y is identical to the marginal distribution of X — knowing Y tells you nothing about X, and every column of the joint table looks the same after normalization. If they are dependent, the shape of the conditional distribution changes as y varies. Comparing f_{X|Y}(x|y) for several values of y shows exactly how Y "informs" X: which values of X become more or less likely as Y shifts. This comparison is the right way to think about dependence, and it directly motivates conditional expectation — the expected value of X given Y = y, which averages the conditional distribution and becomes the central tool in regression and Bayesian inference.