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Lesson 10 of 12 · 14 min

Joint, marginal and conditional distributions as information arrives

A joint distribution describes two variables together; summing or integrating out one variable gives the other's marginal distribution, and conditioning on what has already happened updates both the expected value and the variance.

In short

  • The joint CDF F12(x1,x2)=P(X1≤x1,X2≤x2)F_{12}(x_1, x_2) = P(X_1 \le x_1, X_2 \le x_2) is the probability that both variables are at or below their thresholds.
  • For independent variables the joint PDF is the product of the individual PDFs, and the joint CDF the product of the CDFs.
  • A marginal distribution isolates one variable: integrate (continuous) or sum (discrete) the joint distribution over the other. In a contingency table the marginals are the row and column totals.
  • Conditional variance has the same shortcut as before: Var(X∣Y)=E[X2∣Y]−(E[X∣Y])2\text{Var}(X \mid Y) = E[X^2 \mid Y] - (E[X \mid Y])^2. Conditional covariance measures co-movement within a scenario.
  • In a multi-period price tree, each observed price is a new condition: the expected value moves towards the path taken and the variance shrinks as fewer periods remain.

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Joint, marginal and conditional distributions as information arrives · Statistical Distributions for Financial Asset Prices and Returns