Lesson 11 of 14 · All lessons in this module
  1. Random variables: PMF, PDF and CDF
  2. Expected value, variance and standard deviationVideo lesson
  3. Variance and covariance shortcuts, and the law of large numbers
  4. Discrete distributions: uniform, binomial and Poisson
  5. Continuous uniform and normal distributions, and simulating random draws
  6. Log-normal and logistic distributions, and the moments of key distributions
  7. Integrated cases: reading probabilities and moments from a return model
  8. Conditional probability and probability trees
  9. Conditional expected values and variancesVideo lesson
  10. Covariance from a joint probability function
  11. Joint, marginal and conditional distributions as information arrives
  12. Bayes' formula: updating a probability with new informationVideo lesson
  13. Bayes in practice: diffuse priors, missing inputs and screening tests
  14. Integrated cases: a tree with model forecasts and a Bayes update

This lesson is part of the 2027 curriculum. You are following the 2026 curriculum, where it is not taught in this form. Switch if you are sitting the exam under the 2027 curriculum.

Lesson 11 of 14 · 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.

Unlock this lesson free for 7 days

Create a free account to get 7 days free: every lesson, video and flashcard, and the question bank. No card needed.

Start your 7-day free trialLog in