Statistical Distributions for Financial Asset Prices and ReturnsLocked: included in All Access

Random variables and their distributions: unconditional and conditional expected values, variances and covariances, joint distributions, the key distributions used in finance, and updating beliefs with Bayes' formula.

1. 1. Random variables: PMF, PDF and CDFA random variable is fully described by its distribution: a PMF or PDF shows where the probability sits, and the CDF adds it up from the left.Locked: included in All Access13 min
2. 2. Expected value, variance and standard deviationExpected value is your probability-weighted forecast; variance and standard deviation measure how far the actual outcome is likely to land from that forecast.Video · 7 minLocked: included in All Access14 min
3. 3. Variance and covariance shortcuts, and the law of large numbersVariance is the mean of the square minus the square of the mean, covariance is the mean of the product minus the product of the means, and with enough random observations a sample mean settles on the true mean.Locked: included in All Access14 min
4. 4. Discrete distributions: uniform, binomial and PoissonThree discrete distributions do most of the work in finance: the uniform for equally likely outcomes, the binomial for counting successes in a fixed number of yes/no trials, and the Poisson for counting rare events in a period.Locked: included in All Access15 min
5. 5. Continuous uniform and normal distributions, and simulating random drawsThe continuous uniform spreads probability evenly over an interval and is the raw material of simulation; the normal is the symmetric bell that finance uses as its default model of returns.Locked: included in All Access14 min
6. 6. Log-normal and logistic distributions, and the moments of key distributionsPrices are modelled as log-normal because they cannot fall below zero and their log returns add up over time; the logistic looks like a normal with fatter tails and drives binary-outcome models.Locked: included in All Access15 min
7. 7. Conditional probability and probability treesA probability tree splits an uncertain outcome into scenarios and then into outcomes conditional on each scenario, so joint and total probabilities can be read straight off its branches.Locked: included in All Access14 min
8. 8. Conditional expected values and variancesWork out the expected value inside each scenario, then weight those conditional expectations by the scenario probabilities: the result must match the overall forecast.Video · 6 minLocked: included in All Access15 min
9. 9. Covariance from a joint probability functionWhen returns are forecast with scenarios, covariance is the probability-weighted sum of the cross-products of each asset's deviation from its expected return.Locked: included in All Access14 min
10. 10. Joint, marginal and conditional distributions as information arrivesA 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.Locked: included in All Access14 min
11. 11. Bayes' formula: updating a probability with new informationBayes' formula turns a prior probability into a posterior probability by multiplying it by how much more, or less, likely the new information is when the event is true.Video · 5 minLocked: included in All Access15 min
12. 12. Bayes in practice: diffuse priors, missing inputs and screening testsExam Bayes problems usually hide one input: you may need to back it out with the total probability rule, use complements, or start from equal (diffuse) priors.Locked: included in All Access14 min