Source: Walpole Ch. 5.2, Ch. 6, Lecture Weeks 7–9
Tags: binomial distribution, Poisson distribution, uniform distribution, normal distribution, standard normal, z-score, z-table, STAT 3021, UMN
Difficulty: Intermediate | Prerequisites: Random Variables, Expectation and Variance (Ch. 3-4 notes), basic integration
Chapters 5 and 6 name the distributions that appear everywhere in applied statistics. Instead of working from a raw PMF or PDF each time, you learn to recognise when a situation fits a known distribution and then use its pre-derived formulas for mean, variance, and probabilities. The binomial covers "how many successes in n trials," the Poisson covers "how many events in a given interval," and the normal distribution is the single most important distribution in all of statistics because of the central limit theorem. Mastering the normal distribution and z-tables is essential for the entire second half of the course.
The binomial distribution counts successes in a fixed number of independent trials. The Poisson approximates counts of rare events. The uniform distributes probability evenly across an interval. The normal (bell curve) is symmetric about its mean and is characterised entirely by μ and σ. You convert any normal to the standard normal (Z) using the z-score formula, then use a z-table to find probabilities.
Binomial distribution, X ~ Bin(n, p)
The distribution of the number of successes in n independent trials, each with success probability p. Think of it as answering: "Out of n attempts, how many times does the thing happen?"
Bernoulli trial
A single experiment with exactly two outcomes: success (probability p) or failure (probability 1 - p). A binomial RV is the sum of n independent Bernoulli trials.
Poisson distribution, X ~ Poisson(λ)
A distribution for the count of events occurring in a fixed interval of time or space, where events happen independently at a constant average rate λ. Think of it as: "How many calls arrive at a help desk in one hour?"
Uniform distribution, X ~ Uniform(a, b)
A continuous distribution where every value in the interval [a, b] is equally likely. The PDF is a flat horizontal line. Think of it as perfect randomness across an interval.
Normal distribution, X ~ N(μ, σ²)
The bell-shaped curve, symmetric about its mean μ, with spread controlled by σ. Roughly 68% of values lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ (the 68-95-99.7 rule).
Standard normal distribution, Z ~ N(0, 1)
A normal distribution with mean 0 and standard deviation 1. Every normal distribution can be converted to this one using the z-score.
Z-score (standard score)
Z = (X - μ) / σ. It tells you how many standard deviations X is from the mean. In simple terms, it is a universal unit that lets you compare values from different normal distributions.
Conditions: fixed number of trials n, each trial independent, same probability of success p, two outcomes per trial
PMF: P(X = x) = C(n, x) × pˣ × (1 - p)ⁿ⁻ˣ for x = 0, 1, ..., n
Mean: μ = np
Variance: σ² = np(1 - p)
Example: flip a fair coin 10 times. X = number of heads. X ~ Bin(10, 0.5). P(X = 7) = C(10,7) × 0.5⁷ × 0.5³
Models the count of events in a fixed interval when events occur independently at rate λ
PMF: P(X = x) = e⁻λ × λˣ / x! for x = 0, 1, 2, ...
Mean = λ, Variance = λ (mean and variance are equal)
Can approximate the binomial when n is large and p is small: use λ = np
Example: a call centre receives an average of 4 calls per minute. P(exactly 6 calls in one minute) = e⁻⁴ × 4⁶ / 6!
PDF: f(x) = 1/(b - a) for a ≤ x ≤ b, and 0 otherwise
CDF: F(x) = (x - a)/(b - a) for a ≤ x ≤ b
Mean: μ = (a + b)/2
Variance: σ² = (b - a)²/12
Every sub-interval of equal width has equal probability
PDF: f(x) = (1 / (σ√(2π))) × e^(-(x-μ)²/(2σ²))
Symmetric, bell-shaped, centred at μ
Completely determined by two parameters: μ (centre) and σ (spread)
68-95-99.7 rule (empirical rule):
About 68% of data within μ ± 1σ
About 95% within μ ± 2σ
About 99.7% within μ ± 3σ
Convert X ~ N(μ, σ²) to Z ~ N(0, 1) using Z = (X - μ) / σ
Z-table gives P(Z ≤ z), the area to the left of z
To find P(Z > z): use 1 - P(Z ≤ z)
To find P(a < Z < b): P(Z ≤ b) - P(Z ≤ a)
Reverse lookup: given a probability, find the z-value (used later for confidence intervals)
In R: pnorm(z) gives P(Z ≤ z), qnorm(p) gives the z-value for cumulative probability p
P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \quad \text{(Binomial)}P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \quad \text{(Poisson)}Z = \frac{X - \mu}{\sigma}Students try to apply the binomial formula when trials are not independent (e.g. drawing without replacement from a small population). If the population is large relative to the sample, it is approximately fine, but for small populations you need the hypergeometric distribution.
Confusing the parameters of the normal distribution: N(μ, σ²) uses variance, not standard deviation. Some textbooks write N(μ, σ). Walpole uses σ².
Forgetting to standardise before using the z-table. You cannot look up a raw X value in the z-table; you must convert to Z first.
Students sometimes think the normal distribution applies to everything. It does not. It is symmetric, so it is a poor model for heavily skewed data.
⚠️ Know the conditions for the binomial distribution cold. If a problem describes fixed n, independent trials, constant p, and two outcomes, it is binomial.
⚠️ Z-table calculations are tested on every exam. Practise converting between X-values and Z-scores and reading the table in both directions.
⚠️ The 68-95-99.7 rule is a quick mental check. If an exam answer falls outside these rough bounds, re-examine your calculation.
True or False: For a Poisson distribution, the mean always equals the variance. (True.)
Fill in the blank: The z-score for a value that is 1.5 standard deviations above the mean is ___. (1.5)
True or False: P(Z > 0) = 0.5 for the standard normal. (True, by symmetry.)
Fill in the blank: If X ~ Bin(20, 0.3), then E(X) = ___. (6)
Q: If X ~ Bin(15, 0.4), find P(X = 6).
A: P(X = 6) = C(15, 6) × 0.4⁶ × 0.6⁹ = 5005 × 0.004096 × 0.010077 ≈ 0.2066.
Q: Calls arrive at a rate of 3 per hour. What is the probability of receiving exactly 5 calls in one hour?
A: X ~ Poisson(3). P(X = 5) = e⁻³ × 3⁵ / 5! = 0.0498 × 243 / 120 ≈ 0.1008.
Q: Scores on a test are normally distributed with μ = 72 and σ = 8. What proportion of students scored above 88?
A: Z = (88 - 72) / 8 = 2.0. P(Z > 2.0) = 1 - P(Z ≤ 2.0) = 1 - 0.9772 = 0.0228, or about 2.3%.
Q: For X ~ Uniform(2, 10), find P(4 < X < 7).
A: P(4 < X < 7) = (7 - 4) / (10 - 2) = 3/8 = 0.375.
The normal distribution is the foundation of the central limit theorem (Chapter 8), which explains why sample means are approximately normal regardless of the population shape. Z-scores reappear in confidence intervals and hypothesis tests (Chapters 9-10). The Poisson distribution connects to exponential waiting times, and the binomial connects to the proportion-based inference you see in Chapter 10.
binomial distribution, Bernoulli trial, n choose x, Poisson distribution, lambda, uniform distribution, normal distribution, bell curve, Gaussian, standard normal, z-score, z-table, standardisation, 68-95-99.7 rule, empirical rule, pnorm, qnorm, STAT 3021, Walpole Chapter 5, Walpole Chapter 6