ECMT1010 · Introduction To Economic Statistics
Random Variables & Distributions
Week 12 turns probability into numbers attached to outcomes. It covers the discrete pmf, the expected value E[X] = Σxᵢpᵢ and its rules, the variance and SD of a random variable (Var(X) = E[X²] − μ²), the binomial distribution Bin(n, p) with mean np and SD √(np(1−p)), and continuous RVs where probability is area under a pdf. It is examined as short-answer: build a distribution, compute E[X] and σ, use the binomial, and read a probability as an area.
What this chapter covers
- 011. Discrete random variable and its pmf p(x), with Σp(x) = 1
- 022. Expected value E[X] = Σxᵢpᵢ as a probability-weighted average (the mean μ_X)
- 033. EV rules: E[X + Y] = E[X] + E[Y], E[bX] = b·E[X], E[b] = b, and E[g(X)] = Σg(xᵢ)pᵢ
- 044. Variance Var(X) = E[(X − μ)²] = E[X²] − μ², and SD σ_X = √Var(X)
- 055. The binomial distribution P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ
- 066. Binomial mean np and SD √(np(1 − p))
- 077. Continuous RV and the pdf: total area = 1, probabilities are areas
- 088. The uniform distribution and reading a probability as a geometric area
Expected value and SD of a discrete random variable
- 3 marks(a) Let Pro count = U → Basic = 2U, Plus = 3U; total = 6U. So P(X = 8) = 2/6 = 1/3, P(X = 14) = 3/6 = 1/2, P(X = 24) = 1/6.
- 2 marks(b) E[X] = 8(1/3) + 14(1/2) + 24(1/6) = 2.667 + 7.000 + 4.000 = $13.67.
- 2 marks(c) Variance Var(X) = Σ(xᵢ − μ)²pᵢ = (8 − 13.67)²(1/3) + (14 − 13.67)²(1/2) + (24 − 13.67)²(1/6) = (32.15)(0.3333) + (0.109)(0.5) + (106.7)(0.1667) = 10.72 + 0.055 + 17.79 = 28.56.
- 1 markTake the square root for the SD: σ = √28.56 ≈ $5.34.
Key terms
- Probability mass function (pmf)
- For a discrete random variable, the function p(x) giving the probability of each possible value, with all the probabilities summing to 1. It is the full description of the variable's distribution.
- Expected value E[X]
- The probability-weighted average of a random variable, E[X] = Σxᵢpᵢ = μ_X. It is the long-run mean and obeys the linear rules E[X + Y] = E[X] + E[Y] and E[bX] = b·E[X].
- Variance and SD of a random variable
- Var(X) = E[(X − μ)²] = E[X²] − μ² measures spread around the mean; the standard deviation is σ_X = √Var(X). The shortcut E[X²] − μ² is usually the quickest route.
- Binomial distribution
- The distribution of the number of successes in n independent trials each with success probability p: P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ. Its mean is np and its SD is √(np(1 − p)).
- Probability density function (pdf)
- For a continuous random variable, a curve f(x) whose total area is 1, where probabilities are areas under the curve over an interval rather than heights. P(a < X < b) is the area between a and b.
- Uniform distribution
- A continuous distribution that is flat over an interval [a, b], with f(x) = 1/(b − a), mean (a + b)/2 and variance (b − a)²/12. Probabilities are simple rectangular areas.
Random Variables & Distributions FAQ
What is the difference between expected value and a plain average?
A plain average treats every value equally; expected value weights each possible value by its probability: E[X] = Σxᵢpᵢ. If the outcomes are equally likely the two coincide, but when some values are more probable than others — like a pricing tier with many more cheap subscribers than expensive ones — the expected value is pulled toward the high-probability values. It is the long-run average you would see over many repetitions, not the simple midpoint of the possible values.
When should I use the binomial distribution?
Use the binomial when you are counting the number of 'successes' in a fixed number n of independent trials that each have the same success probability p, and you have two outcomes per trial (success/failure). The probability of exactly k successes is C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ, the mean is np and the SD is √(np(1 − p)). The independence and constant-p conditions are what you should check before reaching for the formula.
How do I compute the variance of a random variable quickly?
The definition is Var(X) = Σ(xᵢ − μ)²pᵢ, but the shortcut Var(X) = E[X²] − μ² is usually faster: first compute μ = E[X], then E[X²] = Σxᵢ²pᵢ, and subtract μ². The SD is the square root. The shortcut avoids carrying a messy μ through every bracket and is less error-prone when the mean is not a round number. Either way, remember the SD has the same units as X, but the variance has squared units.
How does probability work for a continuous random variable?
For a continuous variable, probability is the AREA under the density curve f(x) over an interval, not the height of the curve. The total area is always 1, and the probability of any single exact value is 0 — only intervals have positive probability. For simple shapes like a uniform (rectangle) or a triangular density you can find probabilities with basic geometry: P(a < X < b) is just the area of the region between a and b under the curve.
Exam move
Drill the discrete toolkit until it is mechanical: confirm the probabilities sum to 1, compute E[X] as a probability-weighted sum, then get the variance — prefer the E[X²] − μ² shortcut — and square-root for the SD. For the binomial, learn to recognise the 'fixed n independent trials, two outcomes, constant p' signature and have C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ plus np and √(np(1 − p)) ready. For continuous variables, switch your mindset from heights to areas: total area is 1 and probabilities are regions, so practise the uniform and triangular cases with simple geometry (½·base·height for a triangle). Always attach units to E[X] and σ and finish with a one-line interpretation, which is where the conceptual marks sit.
