Difficulty: Intermediate | Prerequisites: Chapters 1–4 study notes
Chapter 5 moves from raw data to probability models: instead of summarising values you already have, you describe the behaviour of a random variable with its expected value and variance. Chapter 6 extends this to continuous random variables, where probabilities come from areas under a density curve. The two named distributions to know cold are the Poisson (discrete, counts of rare events) and the uniform and exponential (continuous).
Discrete random variable
A variable whose possible values are countable (0, 1, 2, 3, ...). Each value has an assigned probability, and the probabilities must sum to 1.
Expected value – E(X) or μ
The long-run average of a random variable. For a discrete RV: E(X) = Σ[x × P(x)]. Think of it as the "centre of gravity" of the probability distribution.
Variance – Var(X) or σ²
The expected squared deviation from the mean. Measures how spread out the distribution is. Var(X) = E[(X – μ)²] = E(X²) – [E(X)]².
Standard deviation – σ
The square root of the variance. Same units as X, which makes it more interpretable than variance.
Poisson distribution
A discrete distribution for the count of events in a fixed interval (time, area, volume) when events occur independently at a constant average rate λ. Both the mean and the variance equal λ.
Density function – f(x)
For a continuous random variable, f(x) is a curve such that probabilities are areas under it. f(x) itself is not a probability; it can exceed 1. The total area under the curve equals 1.
Cumulative distribution function (CDF) – F(x)
The probability that the random variable takes a value less than or equal to x. F(x) = P(X ≤ x).
Uniform distribution
A continuous distribution where every value in the interval [a, b] is equally likely. The density is flat at 1/(b – a).
Exponential distribution
A continuous distribution that models the time between events in a Poisson process. Its parameter is λ (the rate), and both its mean and standard deviation equal 1/λ.
Percentile
The value below which a given percentage of the distribution falls. The pth percentile satisfies: the integral of f(x) from –∞ to that value equals p.
Expected value (mean)
E(X) = μ = Σ[x × p(x)]
Multiply each possible value by its probability, then sum
Rules for expected values
E(a + bX) = a + bE(X) (constants slide through)
E(X ± Y) = E(X) ± E(Y)
E(g(X)) = Σ[g(x) × p(x)] for any function g
Variance
Var(X) = σ² = E[(X – μ)²] = E(X²) – [E(X)]²
The shortcut form E(X²) – [E(X)]² is usually faster to compute
Rules for variance
Var(a + bX) = b² × Var(X) (additive constants vanish, multiplicative constants get squared)
Var(X ± Y) = Var(X) + Var(Y) (variances always add, regardless of the sign, when X and Y are independent)
Standard deviation = √Var(X)
Models the count of events in a fixed interval when events are independent and occur at a constant rate λ
P(X = x) = (e^–λ × λ^x) / x!
Mean: E(X) = λ
Standard deviation: σ = √λ
Common uses: number of emails per hour, number of typos per page, number of cars passing a point per minute
Density function f(x)
f(x) ≥ 0 for all x
Total area under the curve = 1
P(a ≤ X ≤ b) = integral of f(x) from a to b
Key calculations from a density function
Proportion between a and b: integrate f(x) from a to b
pth percentile: find the value y where the integral from –∞ to y equals p
Median: the value where the integral from –∞ to that point equals 0.5
Mean: E(X) = μ = integral of x × f(x) from –∞ to ∞
E(g(X)) = integral of g(x) × f(x) dx
f(x) = 1/(b – a) for a ≤ x ≤ b, and 0 otherwise
Mean: E(X) = (a + b) / 2
Standard deviation: σ = √[(b – a)² / 12]
Probabilities are just lengths of intervals divided by (b – a)
f(x) = λe^–λx for x ≥ 0, and 0 otherwise
CDF: F(x) = 1 – e^–λx for x ≥ 0
Mean: E(X) = 1/λ
Standard deviation: σ = 1/λ
The "memoryless" property: P(X > s + t | X > s) = P(X > t). The probability of lasting another t units does not depend on how long you have already waited.
Topic | Formula |
|---|---|
Discrete mean | E(X) = Σ[x × p(x)] |
Mean linear rule | E(a + bX) = a + bE(X) |
Mean sum/difference | E(X ± Y) = E(X) ± E(Y) |
Variance | Var(X) = E(X²) – [E(X)]² |
Variance linear rule | Var(a + bX) = b² × Var(X) |
Variance sum/difference | Var(X ± Y) = Var(X) + Var(Y) |
Standard deviation | σ = √Var(X) |
Poisson PMF | P(X = x) = (e^–λ × λ^x) / x! |
Poisson mean and SD | E(X) = λ, σ = √λ |
Uniform density | f(x) = 1/(b – a) on [a, b] |
Uniform mean | (a + b) / 2 |
Uniform SD | √[(b – a)² / 12] |
Exponential density | f(x) = λe^–λx for x ≥ 0 |
Exponential CDF | F(x) = 1 – e^–λx |
Exponential mean and SD | E(X) = 1/λ, σ = 1/λ |
Continuous mean | E(X) = ∫x × f(x) dx |
Proportion between a and b | ∫ from a to b of f(x) dx |
The Poisson distribution is used wherever you count rare, independent events: the number of calls a help desk receives per hour, the number of defects per metre of fabric, the number of accidents at a junction per year. If you see a count of events with no natural upper bound, Poisson is the model to reach for.
The exponential distribution models waiting times: how long until the next bus arrives, how long a light bulb lasts, how long between customer arrivals. Its memoryless property is both its most distinctive feature and its biggest practical limitation, since many real waiting times are not memoryless.
Students often confuse the variance rules for sums and products. When you add or subtract random variables, variances always add (never subtract). The sign in front of Y does not matter because variance measures spread, and spread does not cancel.
For the Poisson distribution, the mean and the variance are the same number (λ). Students sometimes forget this and try to compute them separately.
A density value f(x) is not a probability. It can be greater than 1. Only the area under the curve between two values gives a probability.
In the exponential distribution, the mean is 1/λ, not λ. Students frequently swap these because the Poisson uses λ as the mean directly.
⚠️ Be comfortable computing E(X) and Var(X) from a probability table for a discrete random variable.
⚠️ Know the shortcut variance formula: Var(X) = E(X²) – [E(X)]². Exam problems almost always expect this form.
⚠️ Expect a Poisson problem: you will be given λ and asked for P(X = x) using the PMF formula. Have the factorial values for small numbers ready.
⚠️ For continuous distributions, be prepared to calculate P(a < X < b) by integrating f(x), or by using the CDF shortcut for the exponential: P(X > x) = e^–λx.
⚠️ Know the uniform distribution formulas for mean and standard deviation; these are quick marks.
True or false: Var(X – Y) = Var(X) – Var(Y).
Fill in the blank: For a Poisson distribution, E(X) = ___ and σ = ___.
True or false: A density function value f(x) can never exceed 1.
The exponential distribution with λ = 2 has mean ___ and SD ___.
True or false: Var(3 + 5X) = 25 × Var(X).
Answers: 1. False (variances add: Var(X) + Var(Y)). 2. λ and √λ. 3. False (f(x) can exceed 1; only areas are probabilities). 4. 0.5 and 0.5. 5. True.
Q: X has the following distribution: P(X = 0) = 0.2, P(X = 1) = 0.5, P(X = 2) = 0.3. Find E(X) and Var(X).
A: E(X) = 0(0.2) + 1(0.5) + 2(0.3) = 0 + 0.5 + 0.6 = 1.1. E(X²) = 0²(0.2) + 1²(0.5) + 2²(0.3) = 0 + 0.5 + 1.2 = 1.7. Var(X) = 1.7 – (1.1)² = 1.7 – 1.21 = 0.49.
Q: A call centre receives an average of 3 calls per minute. What is the probability of receiving exactly 5 calls in a given minute?
A: Poisson with λ = 3. P(X = 5) = (e^–3 × 3⁵) / 5! = (0.0498 × 243) / 120 = 12.0954 / 120 ≈ 0.1008.
Q: X is uniformly distributed on [2, 10]. Find the mean, standard deviation, and P(4 < X < 7).
A: Mean = (2 + 10) / 2 = 6. SD = √[(10 – 2)² / 12] = √(64/12) = √5.333 ≈ 2.309. P(4 < X < 7) = (7 – 4) / (10 – 2) = 3/8 = 0.375.
Q: The time between arrivals at a shop follows an exponential distribution with λ = 0.5 per minute. What is the probability of waiting more than 4 minutes?
A: P(X > 4) = e^–λx = e^–(0.5)(4) = e^–2 ≈ 0.1353.
The expected value and variance rules from Ch. 5 generalise naturally to the normal distribution, which is the centrepiece of the inferential statistics you will meet after the midterm. The Poisson and exponential distributions are a matched pair: the Poisson counts events in an interval while the exponential measures the time between those events. Understanding both is essential for modelling real processes.
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