Random Variables and Probability Mass Functions introduced the Bernoulli, Binomial, Geometric, and Poisson PMFs by name; Expectation, Mean, and Variance and Functions of Random Variables built the machinery to compute moments from any PMF. This sub-note carries the algebra to its conclusion: closed-form mean and variance for all four named distributions, derived from first principles rather than quoted. The Binomial derivation is the pedagogical centrepiece — writing it as a sum of independent Bernoulli indicators turns an intractable-looking direct sum into two lines, and is the template used throughout the vault whenever a count variable can be decomposed into simpler pieces.
Learning objectives. By the end of this note you should be able to:
- derive and for a Bernoulli random variable directly from the definition;
- decompose a Binomial random variable as a sum of independent Bernoulli indicators and use linearity (and independence) to get its mean and variance in two lines;
- derive the Geometric mean and variance using a self-similarity ("memorylessness") trick;
- derive the Poisson mean and variance directly from the exponential series.
In this lesson
- Bernoulli
- Binomial — the sum-of-indicators trick
- Geometric — the self-similarity trick
- Poisson — direct summation via the exponential series
Before you start
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