A stock price, a portfolio return, an interest rate — none of these live on a countable grid of values the way Discrete Random Variables do. This topic extends every tool from the previous topic to random variables with a continuum of possible values: the probability density function (PDF) replaces the PMF, the cumulative distribution function (CDF) unifies the discrete and continuous cases under one object, and expectation, conditioning, and Bayes' rule all carry over with sums replaced by integrals. The topic centres on the normal distribution — by far the most important continuous distribution in quantitative finance — building its properties from the ground up, deriving its normalising constant from scratch in a dedicated derivation sub-note, and closing with the multivariate normal, the distribution underlying essentially every linear model of jointly normal asset returns.
By the end of this topic you should be able to:
- verify a candidate PDF is legitimate and compute event probabilities, expectation, and variance from it;
- relate the PDF and PMF through the unifying CDF, and use the CDF to derive the PMF/PDF of a maximum of independent random variables;
- state the normal distribution's PDF, standardise it to compute probabilities via the standard normal table, and derive its normalising constant and moments from first principles;
- build and read a joint density, recover marginals and conditional densities, and apply the continuous Bayes' rule to infer a hidden quantity from a noisy continuous observation;
- state and use the multivariate normal distribution's defining properties.
In this lesson
- Contents
- The big picture
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