Difficulty: Intermediate | Prerequisites: Chapters 1 to 3 study notes
Chapters 4 through 6 move from describing data you already have to modelling uncertainty. Chapter 4 introduces the rules of probability. Chapter 5 covers discrete random variables and two named distributions (binomial and Poisson). Chapter 6 extends the same ideas to continuous random variables, including the uniform and exponential distributions. These chapters are the bridge between descriptive statistics and inference: you need to be comfortable with expected values, variances, and distribution formulas before tackling confidence intervals and hypothesis tests.
Probability measures how likely an event is, with rules for combining events (addition, multiplication, conditional). Random variables assign numbers to outcomes; their behaviour is described by a mean (expected value) and variance. Binomial and Poisson handle discrete counts, while uniform and exponential handle continuous measurements.
Sample space (S)
The set of all possible outcomes of a random experiment.
Think of it as the complete menu of everything that could happen.
P(A) -- probability of event A
The number of times the event occurs divided by the total number of possible outcomes. Always between 0 and 1.
Independent events
Two events A and B are independent if knowing one occurred does not change the probability of the other. Formally: P(A|B) = P(A).
In simple terms, one event has no influence on the other.
Conditional probability, P(A|B)
The probability of A given that B has already occurred. Calculated as P(A and B) divided by P(B).
Think of it as narrowing the sample space to only the outcomes where B happened, then checking how often A also happens.
Discrete random variable
A variable that takes countable values (0, 1, 2, ...). Each value has an associated probability.
Expected value, E(X)
The long-run average of a random variable. For a discrete RV: the sum of each value times its probability.
Think of it as what you would get on average if you repeated the experiment many times.
Variance, Var(X)
A measure of how spread out the values of a random variable are around the expected value.
Binomial distribution
Models the number of successes in n independent trials, each with the same probability of success p.
In simple terms, it counts how many times something happens in a fixed number of attempts.
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 (lambda).
Think of it as counting rare events: emails per hour, accidents per month.
Continuous random variable
A variable that can take any value in an interval. Probabilities are found by integrating the density function.
Probability density function, f(x)
A function whose integral over an interval gives the probability that the continuous RV falls in that interval. The total area under f(x) equals 1.
Uniform distribution
A continuous distribution where every value in the interval [a, b] is equally likely. The density is constant at 1/(b minus a).
Exponential distribution
A continuous distribution that models the time between events in a Poisson process. Governed by a rate parameter lambda.
0 <= P(A) <= 1 for any event A
P(A) = sum of probabilities of its individual outcomes
P(S) = 1 (something in the sample space must happen)
P(empty set) = 0
A and B are independent if and only if all three hold:
P(A|B) = P(A)
P(B|A) = P(B)
P(A and B) = P(A) * P(B) (the multiplication rule for independent events; extends to more than two events)
P(A|B) = P(A and B) / P(B)
General multiplication rule: P(A and B) = P(A) * P(B|A) = P(B) * P(A|B)
For three events: P(A and B and C) = P(A) * P(B|A) * P(C|A and B)
Mean: E(X) = sum of x * p(x)
Rules for means: E(a + bX) = a + bE(X); E(X +/- Y) = E(X) +/- E(Y); E(g(X)) = sum of g(x) * p(x)
Variance: Var(X) = E[(X minus mu)^2] = E(X^2) minus [E(X)]^2
Standard deviation: sigma_X = sqrt(Var(X))
Rules for variance: Var(a + bX) = b^2 * Var(X); if X and Y are independent, Var(X +/- Y) = Var(X) + Var(Y)
Conditions: fixed number of trials n, each trial has two outcomes (success/failure), constant probability p, trials are independent.
P(X = x) = C(n, x) * p^x * (1 minus p)^(n minus x), for x = 0, 1, 2, ..., n
Mean: E(X) = np
Standard deviation: sigma = sqrt(np(1 minus p))
Skewness rule: p < 0.5 means right-skewed, p = 0.5 means symmetric, p > 0.5 means left-skewed.
P(X = x) = (e^(-lambda) * lambda^x) / x!, for x = 0, 1, 2, ...
Mean: E(X) = lambda
Standard deviation: sigma = sqrt(lambda)
E(g(X)) = integral from negative infinity to positive infinity of g(x) * f(x) dx
The proportion of values between a and b = integral from a to b of f(x) dx
Percentile: integral from negative infinity to p of f(x) dx = the percentile value
Median: integral from negative infinity to median of f(x) dx = 0.5
Mean: E(X) = integral from negative infinity to positive infinity of x * f(x) dx
f(x) = 1/(b minus a) for a <= x < b, and 0 otherwise
Mean: E(X) = (a + b) / 2
Standard deviation: sigma = sqrt((b minus a)^2 / 12)
f(x) = lambda * e^(-lambda * x) for x >= 0, and 0 otherwise
CDF: F(x) = 1 minus e^(-lambda * x) for x >= 0
Mean: E(X) = 1/lambda
Standard deviation: sigma = 1/lambda
P(A|B) = \frac{P(A \cap B)}{P(B)}P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)E(X) = \mu_X = \sum x \cdot p(x)\text{Var}(X) = \sigma_X^2 = E(X^2) - [E(X)]^2P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}E(X) = np, \quad \sigma = \sqrt{np(1-p)}P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}E(X) = \lambda, \quad \sigma = \sqrt{\lambda}f(x) = \frac{1}{b-a}, \quad E(X) = \frac{a+b}{2}, \quad \sigma = \sqrt{\frac{(b-a)^2}{12}}f(x) = \lambda e^{-\lambda x}, \quad F(x) = 1 - e^{-\lambda x}E(X) = \frac{1}{\lambda}, \quad \sigma = \frac{1}{\lambda}The binomial distribution models quality-control inspections: if each widget has a 2% defect rate and you inspect 50, X counts the defects. The Poisson distribution is used in call centres to predict arrival rates. The exponential distribution models the time between customer arrivals or between equipment failures.
Students often assume P(A and B) = P(A) * P(B) always. That multiplication shortcut only works when A and B are independent. For dependent events, you must use the general multiplication rule.
Students confuse the mean of a binomial (np) with the probability p itself. np is the expected count, not a probability.
Students forget that for continuous distributions, P(X = exactly some value) = 0. Probabilities come from areas under the curve, not from plugging a single value into f(x).
Students mix up the exponential parameter: if events arrive at rate lambda, the mean wait time is 1/lambda, not lambda.
The conditional probability formula P(A|B) = P(A and B)/P(B) is tested routinely, often with a two-way table.
Be prepared to identify whether a scenario is binomial (fixed n, two outcomes, constant p, independent) or Poisson (counting events in a fixed interval).
Expect a question asking you to compute E(X) and Var(X) for a discrete distribution table.
For the exponential, know both the PDF and the CDF. The CDF (1 minus e^(-lambda * x)) lets you compute probabilities without integration.
True or False: If P(A|B) = P(A), then A and B are independent. (Answer: True.)
Fill in the blank: The mean of a binomial distribution is ______. (Answer: np.)
True or False: For a continuous random variable, P(X = 3) can be greater than zero. (Answer: False, it is always 0.)
Fill in the blank: For an exponential distribution with rate lambda, the mean is ______. (Answer: 1/lambda.)
True or False: A Poisson distribution can model the number of heads in 10 coin flips. (Answer: False, that is binomial.)
Q: You roll a fair die twice. What is the probability of getting a 6 on the first roll and an even number on the second roll? Are these events independent?
A: P(6 on first) = 1/6. P(even on second) = 3/6 = 1/2. The rolls are independent, so P(both) = (1/6)(1/2) = 1/12.
Q: An online shop has a 21% chance that a package arrives on any given day. You order from this shop 10 times in a month. What is the probability that more than one package arrives on a single day?
A: This is binomial with n = 10, p = 0.21. P(X > 1) = 1 minus P(X = 0) minus P(X = 1). P(X = 0) = C(10,0)(0.21)^0(0.79)^10. P(X = 1) = C(10,1)(0.21)^1(0.79)^9. Compute each and subtract from 1.
Q: The time between arrivals at a help desk follows an exponential distribution with a mean of 5 minutes. What is the probability that the next arrival takes more than 8 minutes?
A: Lambda = 1/5 = 0.2. P(X > 8) = 1 minus F(8) = 1 minus (1 minus e^(-0.2 * 8)) = e^(-1.6) = approximately 0.2019.
Q: X is a discrete random variable with E(X) = 4 and Var(X) = 9. Find E(3X + 2) and Var(3X + 2).
A: E(3X + 2) = 3(4) + 2 = 14. Var(3X + 2) = 9 * Var(X) = 9(9) = 81.
The expected value and variance from Chapters 5 and 6 feed directly into the sampling distributions used to build confidence intervals (Chapter 8). The binomial distribution reappears in proportion tests. Understanding how continuous distributions work (area under the curve equals probability) is essential before you tackle the normal distribution and z-scores.
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