Probability Theory 22 | Conditional Expectation (given random variables)

Conditional expectation given a random variable turns “conditioning on an event” into a new random variable that updates uncertainty based on what another variable equals. For discrete random variables, the conditional expectation of X given Y=y is computed by weighting each possible value of X with the conditional probability P(X=x | Y=y), producing a function of y. That function then becomes a random variable by plugging in the actual random value Y(ω), written as E[X | Y]. This matters because it’s the bridge from basic conditional probability to the machinery used in stochastic processes, where conditioning often depends on evolving random states rather than a fixed event.

In the discrete setting, the course first recalls the earlier definition of E[X | B] for an event B with positive probability, where conditioning reduces to a sum over X’s possible outcomes. It then specializes B to the event “Y equals y,” yielding E[X | Y=y]. Using the standard conditional probability formula, the conditional probability P(X=x | Y=y) becomes a ratio involving the joint probability mass function of (X,Y). As y changes, the conditional expectation changes too, so E[X | Y] is naturally viewed as a real-valued function of y, which is then composed with the random variable Y to form a new random variable.

A die example makes the construction concrete. Let X indicate whether the die outcome is even (X=1 for {2,4,6}, otherwise 0). Let Y indicate whether the outcome is 6 (Y=1 only when the die shows 6, otherwise 0). For Y=0, the conditional expectation E[X | Y=0] averages X over the remaining outcomes {1,2,3,4,5}, where X=1 occurs in {2,4}, giving 2/5. For Y=1, conditioning forces the outcome to be 6, so X must be 1 and E[X | Y=1]=1. Together, these values define the random variable E[X | Y], which encodes how knowledge about Y shifts the expected value of X.

The discussion then extends the same idea to absolutely continuous random variables. When X and Y have a joint density f(x,y), the conditional expectation E[X | Y] is computed using an integral: the numerator integrates x times the joint density over x, while the denominator divides by the marginal density of Y. The resulting expression is a function g(y) that again becomes a random variable via composition g(Y).

Finally, several core properties are highlighted. If X and Y are independent, conditioning on Y removes any dependence, so E[X | Y] collapses to the ordinary expectation E[X]. A related simplification appears for products: conditioning on Y of X·Y reduces to E[X]·(a constant in the appropriate sense), because the Y-related terms cancel under the conditioning structure. Also, E[X | X] returns X itself. The course closes with the “law of total expectation,” stating that averaging the conditional expectations reproduces the unconditional expectation: E(E[X | Y]) = E[X]. This ties conditional probability weighting back to the familiar overall mean and sets up the later use of conditional expectations in stochastic processes.