MAST90105 · Methods Of Mathematical Statistics
Discrete Random Variables and MGFs
A discrete random variable is fully described by its probability mass function p(x) = P(X = x), and from that one object everything follows: the CDF F(x) by accumulating mass, the expectation E[X] = ∑x·p(x) as the long-run average, and the variance Var(X) = E[X²] − (E[X])² as the spread. The chapter then introduces the subject’s most powerful bookkeeping device, the moment generating function MX(t) = E[etX]: differentiate it at t = 0 to read off moments, and exploit its uniqueness to name the distribution of a sum — because the MGF of a sum of independents is the product of their MGFs. You meet the standard discrete families — Bernoulli, binomial, geometric, negative binomial, Poisson — each as a story (counting successes, waiting times, rare events) with its mean, variance and MGF. The provided exam table lists those formulas; this chapter teaches you which family the story implies and how the MGF stitches them together.
What this chapter covers
- 012.1 The PMF and the CDF of a discrete variable
- 022.2 Expectation and the law of the unconscious statistician
- 032.3 Variance via E[X²] − (E[X])²
- 042.4 The moment generating function and reading off moments
- 052.5 MGFs of sums — the product rule for independents
- 062.6 The discrete families: Bernoulli, binomial, geometric, Poisson
Worked example: the MGF of a Poisson and its mean & variance
- +1Set up the MGF. MX(t) = E[etX] = ∑x≥0 etx e−μμx/x! = e−μ ∑x≥0 (μet)x/x!.
- +1Recognise the exponential series. ∑x≥0 zx/x! = ez, so MX(t) = e−μeμet = exp[μ(et − 1)].
- +1First derivative at 0: M′(t) = μet·exp[μ(et−1)], so E[X] = M′(0) = μ.
- +1Second moment: M″(0) = μ + μ² (differentiate again), so E[X²] = μ + μ².
- +1Variance: Var(X) = E[X²] − (E[X])² = (μ + μ²) − μ² = μ. Mean and variance both equal μ — the signature of the Poisson.
Key terms
- Probability mass function (PMF)
- p(x) = P(X = x) for a discrete variable: non-negative and summing to one over the support. Every other quantity — CDF, mean, variance, MGF — is computed from it.
- Expectation
- E[X] = ∑x·p(x), the probability-weighted average. By the law of the unconscious statistician, E[g(X)] = ∑g(x)p(x), so you never need the distribution of g(X) to find its mean.
- Variance
- Var(X) = E[(X − μ)²] = E[X²] − (E[X])² — the mean squared deviation, almost always computed from the second moment minus the squared mean. Its square root is the standard deviation.
- Moment generating function (MGF)
- MX(t) = E[etX]. Its k-th derivative at t = 0 gives the k-th raw moment, and because the MGF determines the distribution uniquely, matching MGFs identifies the law of a sum or transformation.
- MGF of a sum
- For independent X and Y, MX+Y(t) = MX(t)MY(t). This product rule is why sums of independent Poissons are Poisson, sums of independent normals are normal, and so on — the cleanest tool for the distribution of a sum.
Discrete Random Variables and MGFs FAQ
Why bother with the MGF when the table gives me the mean and variance?
Two reasons the table cannot replace. First, the MGF identifies the distribution of a sum of independent variables by multiplying MGFs — the fastest way to show a sum of Poissons is Poisson, or a sum of normals is normal. Second, it is a uniqueness tool: if two variables share an MGF they share a distribution. The table gives formulas; the MGF gives derivations and the recipe for sums.
How do I pick which discrete family a word problem implies?
Match the story. A single yes/no trial is Bernoulli; a fixed number of independent trials counting successes is binomial; the number of trials until the first success is geometric; the count of rare events in a fixed interval is Poisson. Once you have named the family, its mean, variance and MGF come straight off the provided table — the marks are in naming it, not memorising the formula.
What is the quickest way to compute a variance?
Use Var(X) = E[X²] − (E[X])². Compute the first and second raw moments — directly from the PMF or by differentiating the MGF twice at t = 0 — then subtract the square of the mean. Working from the definition E[(X−μ)²] is correct but slower and more error-prone.
Exam move
Anchor on the PMF: from it, practise producing the CDF, mean, variance and MGF without looking anything up. Make the MGF a reflex — derive the Poisson and binomial MGFs once by hand so you trust the ‘differentiate at zero’ and ‘product for sums’ moves, since both recur in the transformations chapter. For the named families, learn the stories (success counts, waiting times, rare events) rather than the formulas, because the exam table supplies the formulas and the marks are in choosing the family. Always compute variance as the second moment minus the squared mean.
