Discrete distributions

A probability mass function places pieces of a one-unit probability pie on countable values.

A random variable X maps each outcome ω to a real number X(ω).

For discrete X, p(x)=P(X=x) is nonnegative and its values sum to 1.

The CDF F(x)=P(X≤x) accumulates mass and never decreases.

The underlying outcome can be descriptive, such as which card was drawn, while a random variable records a number such as its points. Different scoring rules on the same experiment create different random variables.

A PMF is a table of exact-value probabilities. Check that every entry is at least zero and that the entire table sums to one before using it for any calculation.

The CDF at x adds every mass at or below x. For a discrete variable it looks like a staircase: flat between possible values and jumping upward by p(x) at each value.

Expected value as a weighted balance point

Expectation is the long-run average produced by weighting every value by its probability.

Multiply each possible value by how often it occurs, then add.

Expectation can be a value the variable never takes; a fair die has mean 3.5.

Linearity E[aX+b]=aE[X]+b holds even without independence.

Expectation is a weighted sum, not an ordinary average of the listed values. A value occurring with probability 0.9 must influence the balance point nine times as strongly as a value occurring with probability 0.1.

Linearity lets us price totals without finding their full joint distribution: E[X+Y]=E[X]+E[Y], even when X and Y depend on each other. Independence becomes important for many variance calculations, not for this expectation identity.

An insurer multiplies each possible claim amount by its probability and adds to obtain expected claim cost, then adds expenses and a risk margin to set premiums. Expectation is useful because it predicts portfolio averages, while it cannot describe tail risk for one customer by itself.

Practical deep dive for this lesson

Turn a real payoff table into a decision

A distribution becomes useful when every possible numerical consequence and its probability are made explicit, checked, and summarized for the decision horizon.

List outcomes, map each to a profit or cost, and combine duplicate numerical values into one PMF entry. Check that probability sums to one before calculating expectation.

Multiply each value by its mass and add to obtain average payoff per repetition. For many independent repetitions, expected totals scale linearly, but actual cash needs can still be driven by variance and worst cases.

Compare expected payoff with fees, risk tolerance, and alternative actions. A positive expectation can coexist with a high chance of loss, so decisions usually need quantiles or variance in addition to the mean.