Source: Purdue University, Introduction to Statistics
Tags: random variable, probability mass function, PMF, expected value, variance, standard deviation, binomial distribution, BInS, Poisson distribution, cumulative distribution function, CDF, continuous random variable, density function, normal distribution, z-table, z-score, normal approximation to binomial, continuity correction, QQ plot, uniform distribution, exponential distribution
Difficulty: Intermediate to Advanced Prerequisites: Chapter 3 (mean, variance concepts) and Chapter 4 (probability rules, independence). You should be comfortable with summation notation and basic integration concepts.
Chapters 5 and 6 move from "here is some data, let us describe it" to "here is a model that generates data." A random variable assigns a number to each outcome in a sample space, and a probability distribution tells you how likely each value is. Chapter 5 covers discrete distributions (values you can list), while Chapter 6 covers continuous distributions (values that fill an interval). The two most important named distributions for this exam are the binomial (discrete, Chapter 5) and the normal (continuous, Chapter 6). Mastering these is essential, because sampling distributions in Chapter 7 and all future inference lean on the normal distribution.
A discrete random variable has a probability mass function that sums to 1; a continuous random variable has a density function that integrates to 1. The binomial, Poisson, uniform, exponential, and normal distributions each have their own formula for probabilities, means, and standard deviations. The normal distribution is the workhorse of statistics, and you need to be fluent with z-table lookups.
Random variable (X)
A function that assigns a numerical value to each outcome in a sample space.
Think of it as a rule that turns experimental outcomes into numbers.
Probability mass function (PMF)
For a discrete random variable, p(x) = P(X = x). It gives the probability for each possible value.
Valid probability distribution (discrete)
Two conditions: (a) 0 ≤ p(x) ≤ 1 for every x, and (b) Σ p(x) = 1.
Expected value (mean) of a discrete random variable
E(X) = μ_X = Σ x · p(x). It is the long-run average value of X.
In simple terms, if you repeated the experiment forever, the average outcome would converge to E(X).
Variance of a discrete random variable
Var(X) = σ²_X = E[(X − μ_X)²] = E(X²) − [E(X)]².
It measures how spread out the distribution is around the mean.
Standard deviation of a discrete random variable
σ_X = √Var(X). Same units as X.
Binomial distribution
Models the number of successes in n independent trials, each with the same probability of success p. Conditions (BInS): Binary outcomes, Independent trials, n is fixed, Same p for every trial.
Poisson distribution
Models the number of events occurring in a fixed interval of time or space, when events happen independently at a constant average rate λ.
Think of it as a count of rare-ish events: emails per hour, accidents per month, typos per page.
Cumulative distribution function (CDF)
F(x) = P(X ≤ x). For discrete variables, it is the running total of the PMF. For continuous variables, it is the integral of the density from −∞ to x.
Probability density function (PDF)
For a continuous random variable, f(x) gives the "height" of the distribution at x. Probabilities come from areas under the curve, not from f(x) itself.
Normal distribution
A continuous, bell-shaped, symmetric distribution defined by its mean μ and standard deviation σ. The standard normal has μ = 0 and σ = 1.
Z-score
z = (x − μ) / σ. It tells you how many standard deviations x is from the mean. Used with the z-table to find probabilities and percentiles.
Continuity correction
An adjustment of ±0.5 applied when using the continuous normal distribution to approximate a discrete distribution (e.g. the binomial). It improves accuracy.
QQ plot (quantile-quantile plot)
A graphical tool that plots the quantiles of your data against the theoretical quantiles of a normal distribution. If the points lie roughly on a straight line, normality is plausible.
Uniform distribution
A continuous distribution where every value in the interval [a, b] is equally likely. f(x) = 1/(b − a) for a ≤ x < b.
Exponential distribution
A continuous distribution that models the time between events in a Poisson process. f(x) = λe^(−λx) for x ≥ 0.
E(X) = Σ x · p(x)
E(g(X)) = Σ g(x) · p(x), which covers computing E(X²) by setting g(x) = x².
Var(X) = E(X²) − [E(X)]²
σ_X = √Var(X)
E(a + bX) = a + bE(X)
E(X ± Y) = E(X) ± E(Y) (always, whether or not X and Y are independent)
Var(a + bX) = b²Var(X). The additive constant a drops out.
If X and Y are independent: Var(X ± Y) = Var(X) + Var(Y). Note the plus sign even for subtraction.
Check BInS: Binary outcomes, Independent trials, n fixed, Same probability p.
P(X = x) = C(n, x) · pˣ · (1 − p)^(n−x), for x = 0, 1, 2, ..., n.
E(X) = np
σ = √[np(1 − p)]
Skewness depends on p:
p < 0.5 → right-skewed
p = 0.5 → symmetric
p > 0.5 → left-skewed
To compute P(X > x), P(X ≥ x), P(X < x), or P(X ≤ x), either sum individual binomial probabilities or use complement relationships: P(X ≥ x) = 1 − P(X ≤ x − 1).
Appropriate when counting independent events over a fixed interval at a constant average rate.
P(X = x) = e^(−λ) · λˣ / x!, for x = 0, 1, 2, ...
E(X) = λ
σ = √λ
Watch the units of λ. If the rate is given per hour but the question asks about 30 minutes, halve λ.
Same complement strategies apply for cumulative probabilities.
F(x) = P(X ≤ x)
For a discrete random variable, F(x) = Σ p(k) for all k ≤ x.
P(a < X < b) = ∫ from a to b of f(x) dx (the area under the curve between a and b).
CDF: F(y) = ∫ from −∞ to y of f(x) dx.
Percentile: find y such that F(y) = p.
Median: find μ̃ such that F(μ̃) = 0.5.
E(X) = ∫ from −∞ to ∞ of x · f(x) dx.
E(g(X)) = ∫ from −∞ to ∞ of g(x) · f(x) dx (used for variance and standard deviation calculations).
Denoted X ~ N(μ, σ).
Symmetric, bell-shaped, centred at μ with spread controlled by σ.
To find probabilities, standardise: z = (x − μ) / σ, then look up the z-value in the z-table.
The z-table gives P(Z ≤ z). For P(Z > z), use 1 − P(Z ≤ z). For P(a < Z < b), compute P(Z ≤ b) − P(Z ≤ a).
To find a percentile, look up the desired probability in the body of the z-table, read off z, then back-transform: x = μ + zσ.
When np ≥ 10 and n(1 − p) ≥ 10, the binomial distribution is well approximated by N(np, √[np(1 − p)]).
Apply the continuity correction (±0.5) because you are approximating a discrete distribution with a continuous one. For example: P(X ≥ 8) becomes P(Y ≥ 7.5) in the normal approximation.
The continuity correction is the default method for this course.
If the points follow a roughly straight line, the data are plausibly normal.
Systematic curvature indicates non-normality:
Points curving upward on the right → right-skewed.
Points curving downward on the right → left-skewed.
An S-shape → symmetric but with heavier or lighter tails than normal.
f(x) = 1/(b − a) for a ≤ x < b, and 0 otherwise.
E(X) = (a + b) / 2
σ = √[(b − a)² / 12]
Probabilities are just proportions of the interval length: P(c < X < d) = (d − c) / (b − a).
f(x) = λe^(−λx) for x ≥ 0, and 0 otherwise.
F(x) = 1 − e^(−λx) for x ≥ 0.
E(X) = 1/λ
σ = 1/λ
To find P(X > x), use 1 − F(x) = e^(−λx). The CDF has a closed form, so no integration by hand is needed.
Discrete expected value: E(X) = Σ x · p(x)
Discrete variance: Var(X) = E(X²) − [E(X)]²
Binomial PMF: P(X = x) = C(n, x) pˣ (1 − p)^(n−x)
Binomial mean and SD: E(X) = np, σ = √[np(1 − p)]
Poisson PMF: P(X = x) = e^(−λ) λˣ / x!
Poisson mean and SD: E(X) = λ, σ = √λ
Normal standardisation: z = (x − μ) / σ
Normal approximation condition: np ≥ 10 and n(1 − p) ≥ 10
Uniform PDF: f(x) = 1/(b − a)
Uniform mean and SD: E(X) = (a + b)/2, σ = (b − a) / √12
Exponential PDF: f(x) = λe^(−λx)
Exponential CDF: F(x) = 1 − e^(−λx)
Exponential mean and SD: E(X) = 1/λ, σ = 1/λ
The binomial distribution is behind every "pass/fail" scenario at scale: quality control on a production line (how many defective items in a batch of 100?), clinical trials (how many patients respond to treatment out of 50?), and election polling (how many respondents prefer candidate A?). The Poisson distribution models rare events: the number of server crashes per month, the number of goals in a football match, the number of calls to a helpline per hour. The exponential distribution models the waiting time between those Poisson events. The normal distribution underpins almost everything in inferential statistics, because the Central Limit Theorem (Chapter 7) guarantees that sample means are approximately normal for large samples, regardless of the underlying distribution.
Students sometimes try to use the binomial formula when the trials are not independent or when p changes from trial to trial. Check all four BInS conditions before assuming a binomial model.
Confusing the Poisson parameter λ with the exponential parameter λ. In the Poisson, λ is the mean count; in the exponential, 1/λ is the mean waiting time. They are reciprocals of each other.
Students forget that for continuous distributions, P(X = any single value) = 0. Probabilities only come from intervals: P(a < X < b).
Skipping the continuity correction in the normal approximation to the binomial. The exam expects you to use it unless told otherwise.
When reading a QQ plot, students sometimes call right-skewed data "left-skewed" or vice versa. Look at which end of the plot departs from the line and in which direction.
⚠️ You will be asked to verify that a probability distribution is valid (all probabilities between 0 and 1, and they sum/integrate to 1).
⚠️ Expect calculation questions for E(X), Var(X), and σ for both generic discrete distributions and named distributions (binomial, Poisson).
⚠️ Identifying whether a scenario is binomial, Poisson, or neither is a likely question. Know the conditions cold.
⚠️ Z-table lookups (both directions: given x find probability, given probability find x) are heavily tested.
⚠️ The normal approximation to the binomial with continuity correction will appear. Know when it is appropriate (np ≥ 10, n(1 − p) ≥ 10) and how to apply the ±0.5 adjustment.
⚠️ QQ plot interpretation (normal, right-skewed, left-skewed, symmetric but non-normal) is tested.
True or False: For a valid discrete distribution, Σ p(x) can equal 0.95.
Fill in the blank: If X ~ Binomial(n = 20, p = 0.3), then E(X) = __________.
True or False: For the Poisson distribution, the mean and variance are both equal to λ.
Fill in the blank: The continuity correction for P(X ≥ 5) when using the normal approximation is P(Y ≥ __________).
True or False: In a QQ plot, a roughly straight line indicates the data are plausibly normal.
Answers: 1. False (must equal 1). 2. 6. 3. True. 4. 4.5. 5. True.
Q: A random variable X has the following distribution: P(X=1) = 0.2, P(X=2) = 0.3, P(X=3) = 0.5. Calculate E(X) and Var(X).
A: E(X) = 1(0.2) + 2(0.3) + 3(0.5) = 0.2 + 0.6 + 1.5 = 2.3. E(X²) = 1(0.2) + 4(0.3) + 9(0.5) = 0.2 + 1.2 + 4.5 = 5.9. Var(X) = 5.9 − (2.3)² = 5.9 − 5.29 = 0.61.
Q: A fair coin is flipped 10 times. What is the probability of getting exactly 6 heads?
A: X ~ Binomial(n = 10, p = 0.5). P(X = 6) = C(10, 6)(0.5)⁶(0.5)⁴ = 210 × (0.5)¹⁰ = 210/1024 ≈ 0.2051.
Q: Calls arrive at a switchboard at an average rate of 4 per hour. What is the probability of receiving exactly 2 calls in a given hour?
A: X ~ Poisson(λ = 4). P(X = 2) = e⁻⁴ · 4² / 2! = e⁻⁴ · 16 / 2 = 8e⁻⁴ ≈ 8(0.0183) ≈ 0.1465.
Q: X ~ N(100, 15). Find P(X < 120).
A: z = (120 − 100) / 15 = 20/15 ≈ 1.33. From the z-table, P(Z < 1.33) ≈ 0.9082.
Q: When is the normal approximation to the binomial appropriate, and what adjustment do you apply?
A: It is appropriate when np ≥ 10 and n(1 − p) ≥ 10. You apply the continuity correction (±0.5) because you are approximating a discrete distribution with a continuous one.
The rules for E(X ± Y) and Var(X ± Y) from Chapter 5 are used in Chapter 7 for the sampling distribution of the sample mean.
The binomial distribution connects back to the multiplication rule for independent events (Chapter 4).
The normal distribution is the centrepiece of Chapter 7 (Central Limit Theorem) and all inferential methods.
The Poisson and exponential distributions are linked: if events follow a Poisson process with rate λ, the waiting times between events follow an Exponential(λ) distribution.
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