Lesson 7 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 7 of 14 · 15 min

Integrated cases: reading probabilities and moments from a return model

To answer a probability question about a normal return, standardise it, read the cumulative probability, and combine the two tails; to answer a question about a price, move to the log-normal and use its exact moments.

In short

  • Standardise first: z=(x−μ)/σz = (x - \mu)/\sigma. Probabilities then come from the standard normal CDF, N(z)N(z), which is symmetric: N(−z)=1−N(z)N(-z) = 1 - N(z).
  • P(a<X<b)=N(zb)−N(za)P(a < X < b) = N(z_b) - N(z_a). A right-tail probability is 1−N(z)1 - N(z).
  • A cut-off return for a given probability works backwards: x=μ+zσx = \mu + z\sigma, with zz = −1.645 for the lowest 5% and −2.33 for the lowest 1%.
  • If ln⁡(P1/P0)\ln(P_1/P_0) is normal with mean μ\mu and standard deviation σ\sigma, the price relative has mean eμ+σ2/2e^{\mu+\sigma^2/2}, median eμe^{\mu}, and positive skew.
  • Exact skewness and excess kurtosis are known for each distribution in the module, and the sign alone is often enough for a concept question.

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