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Lesson 8 of 12 · 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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Conditional expected values and variances · Statistical Distributions for Financial Asset Prices and Returns