QBUS5001 · Foundation In Data Analytics For Business
Random Variables & Discrete Distributions
Module 3 moves from events to random variables: expected value and variance of a discrete distribution, the covariance of two random variables, and the rules for linear transforms and sums that power the portfolio application (where negative correlation reduces risk below the weighted-average SD). It then introduces the two workhorse discrete distributions, the Binomial (fixed trials, count of successes) and the Poisson (counts in a window, with mean equal to variance).
Excel does the heavy lifting with BINOM.DIST and POISSON.DIST, but the exam tests whether you can pick the right model and apply the cumulative-probability relations.
What this chapter covers
- 01Expected value: μ = E[X] = Σ xᵢ p(xᵢ)
- 02Variance: σ² = E(X²) − [E(X)]²
- 03Covariance and correlation of random variables
- 04Linear transforms: E(aX) = aE(X), V(aX) = a²V(X)
- 05Sum of two RVs: V(aX+bY) = a²V(X) + b²V(Y) + 2ab·Cov(X,Y)
- 06Portfolio expected return and risk; diversification
- 07Binomial distribution: P(X=x) = C(n,x)pˣ(1−p)ⁿ⁻ˣ, mean np
- 08Poisson distribution: mean = variance = μ; counts in a window
- 09Cumulative relations: P(X>x) = 1 − P(X≤x)
Portfolio expected return and risk with negative correlation
- 1 markPortfolio expected return: E(R_p) = w_A·E(A) + w_B·E(B) = 0.4(0.09) + 0.6(0.04) = 0.036 + 0.024 = 0.060, i.e. 6.0%.
- 1 markWrite the variance formula: σ_p² = w_A²σ_A² + w_B²σ_B² + 2 w_A w_B ρ σ_A σ_B.
- 2 marksFirst two terms: 0.4²(0.15²) = 0.16×0.0225 = 0.0036; and 0.6²(0.05²) = 0.36×0.0025 = 0.0009.
- 1 markCovariance term: 2(0.4)(0.6)(−0.30)(0.15)(0.05) = 2×0.24×(−0.30)×0.0075 = −0.000324.
- 1 markSum: σ_p² = 0.0036 + 0.0009 − 0.000324 = 0.003276; σ_p = √0.003276 = 0.0572 (5.72%).
- 1 markInterpret: the negative correlation makes the covariance term subtract, pulling portfolio risk (5.72%) below the weighted average of the two SDs — the diversification benefit.
Key terms
- Expected value E[X]
- The probability-weighted average of a random variable, μ = Σ xᵢ p(xᵢ); the long-run mean outcome.
- Binomial distribution
- Models the number of successes in n independent trials each with success probability p: P(X=x) = C(n,x)pˣ(1−p)ⁿ⁻ˣ, with E[X] = np and Var(X) = np(1−p).
- Poisson distribution
- Models the count of events in a fixed window with rate μ: P(X=x) = μˣe⁻ᵁ/x!; uniquely, its mean equals its variance (μ).
- Covariance of random variables
- Cov(X,Y) = E(XY) − E(X)E(Y); the term that appears (doubled and weighted) in the variance of a sum or portfolio.
- Diversification
- The reduction in portfolio risk achieved by combining assets that are not perfectly positively correlated; the lower the correlation, the larger the variance reduction.
Random Variables & Discrete Distributions FAQ
How do I decide between the Binomial and the Poisson?
Use the Binomial when there is a fixed number of trials n and you count successes (e.g. 5 conversions out of 12 emails). Use the Poisson when you count events in a continuous interval of time or space with no fixed upper limit (e.g. chat requests per 30 minutes).
What is the quickest way to get P(X ≥ x) in Excel?
Use the cumulative relation P(X ≥ x) = 1 − P(X ≤ x−1). For a Poisson that is 1 − POISSON.DIST(x−1, μ, 1); for a Binomial it is 1 − BINOM.DIST(x−1, n, p, 1). The cumulative flag (1) is essential.
Why does the covariance term in the portfolio variance have a factor of 2?
Expanding V(w_A X + w_B Y) produces two identical cross terms, w_A w_B Cov(X,Y) twice, which combine into 2 w_A w_B Cov(X,Y). Forgetting the 2 is the most common error in portfolio questions.
Exam move
Memorise the variance-of-a-sum formula as one block and practise it with both positive and negative correlations so the diversification logic becomes intuitive. For the discrete distributions, build a tiny decision rule (fixed trials → Binomial, rate per window → Poisson) and rehearse the Excel functions with the cumulative flag, since exam questions almost always ask for a tail probability rather than a single point.
