Lesson 9 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 9 of 14 · 15 min

Conditional expected values and variances

Work out the expected value inside each scenario, then weight those conditional expectations by the scenario probabilities: the result must match the overall forecast.

In short

  • A conditional expected value E(X∣S)E(X \mid S) averages the outcomes using their probabilities given scenario S.
  • The total probability rule for expected value: E(X)=∑E(X∣Si)P(Si)E(X) = \sum E(X \mid S_i)P(S_i).
  • Conditional and unconditional forecasts must be consistent; if they are not, other investors can profit at your expense.
  • Each scenario has its own conditional variance, measured around E(X∣S)E(X \mid S), which shows the risk within that scenario.
  • When a scenario is revealed, the best forecast moves from E(X)E(X) to E(X∣S)E(X \mid S). This is how analysts update.

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