BUSS1020 · Quantitative Business Analysis
Discrete Probability Distributions
Discrete Probability Distributions (Week 4, Berenson Ch 5) models count-type random variables. You learn to compute the expected value and variance of any discrete distribution, then apply three named models: the binomial (a fixed number of independent yes/no trials), the Poisson (counts of events in a continuous window of opportunity), and the hypergeometric (sampling without replacement from a finite population). Recognising which model fits the situation is the examinable skill.
What this chapter covers
- 01Discrete random variable and its probability distribution
- 02Expected value μ = E(X) = Σ xᵢ P(X = xᵢ)
- 03Variance σ² = E(X²) − [E(X)]² and standard deviation
- 04Rules for combining random variables (E and Var of aX + bY)
- 05Binomial model: fixed n, two outcomes, constant π
- 06Poisson model: events per window, μ = σ² = λ
- 07Hypergeometric model: sampling without replacement
- 08Choosing the right model; Excel functions (BINOM.DIST, POISSON.DIST)
Binomial probability, mean and standard deviation
- 1 markCheck the binomial conditions: fixed n = 15 trials, two outcomes (success/fail), constant π = 0.80, independent trials — all met.
- 1 markExpected number of successes μ = nπ = 15 × 0.80 = 12.
- 1 markVariance σ² = nπ(1 − π) = 15 × 0.80 × 0.20 = 2.4.
- 1 markStandard deviation σ = √2.4 ≈ 1.55 successes.
- 1 markProbability of exactly 13: P(X = 13) = ₁₅C₁₃ (0.80)¹³ (0.20)².
- 1 markExcel: =BINOM.DIST(13, 15, 0.8, 0) for exactly 13 (the 0 requests the PMF, not the cumulative).
Key terms
- Expected value E(X)
- The long-run average of a discrete random variable, E(X) = Σ xᵢ P(X = xᵢ); it is the distribution's mean μ.
- Binomial distribution
- Models the number of successes in a fixed number n of independent trials, each with constant success probability π: P(X = x) = ₙCₓ πˣ(1−π)ⁿ⁻ˣ, with μ = nπ and σ² = nπ(1−π).
- Poisson distribution
- Models the count of events in a continuous interval (time, area, volume) at average rate λ: P(X = x) = e⁻λ λˣ/x!, where the mean equals the variance, μ = σ² = λ.
- Hypergeometric distribution
- Models successes when sampling WITHOUT replacement from a finite population of size N containing A successes; unlike the binomial, trials are not independent.
- Window of opportunity
- The Poisson framing — a fixed continuous span (e.g. a 5-minute window) within which events can occur any number of times with no upper limit.
Discrete Probability Distributions FAQ
How do I decide between binomial, Poisson and hypergeometric?
Binomial: a fixed number of independent yes/no trials with constant probability. Poisson: counts of events occurring in a continuous window with no fixed upper limit. Hypergeometric: sampling a fixed number of items WITHOUT replacement from a finite population. A quick decision tree on 'fixed n? counts in an interval? without replacement?' settles most questions.
Why does the Poisson distribution have its mean equal to its variance?
It is a defining property of the Poisson model: both equal the rate parameter λ. This is also a handy diagnostic — if real count data show a variance much larger than the mean, a plain Poisson model may be a poor fit.
When can I approximate the hypergeometric with a binomial?
When the sample is small relative to the population (a common rule is n less than about 5% of N), sampling without replacement behaves almost like sampling with replacement, so the binomial gives a close, simpler approximation.
Exam move
The high-value skill here is RECOGNITION, not computation. Build a one-line checklist for each model — its conditions, its PMF, its mean and variance, and its Excel function — and practise classifying short scenarios before doing any arithmetic. Memorise the binomial and Poisson mean/variance shortcuts so you can answer those parts instantly. Drill the 'at least' and 'at most' complements, and always confirm the model's assumptions in your written answer, because stating them earns marks even if the numbers are involved.
