25 lessons across 4 weeks, 79 verified past-year questions. Free preview below — sign in for the full interactive lessons, AI tutor, and mock exams.
Week 1
- Joint PMF of Two Discrete Random Variables — Defines the joint probability mass function for two discrete random variables as a table of probabilities over all pairs of values, with properties that entries sum to one.
- Marginal PMF of Discrete Random Variables — Derives the individual (marginal) PMF of each random variable from the joint PMF by summing over all values of the other variable(s), and shows that marginals do not uniquely determine the joint.
- Conditional PMF of One Random Variable Given Another — Defines the conditional PMF of one discrete random variable given a specific value of another, computed as the ratio of joint to marginal PMF, and establishes the product (factorization) rule.
- Examples on Joint, Marginal and Conditional PMFs — Applies the factorization rule through worked examples—including die-and-coin, Poisson-coin (showing Poisson thinning), and IPL powerplay—to build joint PMFs from marginals and conditionals and recover unknown marginals.
- Joint and Marginal PMF of Multiple Discrete Random Variables — Extends joint and marginal PMF definitions to n discrete random variables, showing how to marginalize any subset by summing the joint PMF over all unwanted variables.
- Conditioning and Factorization with Multiple Discrete Random Variables — Generalises conditional PMFs to multiple random variables and shows how any joint PMF can be factored into a product of a marginal and sequential conditional PMFs in any ordering.
Week 2
- Independence of Two Random Variables — Two discrete random variables are independent if and only if their joint PMF equals the product of their marginal PMFs for all values.
- Independence of Multiple Random Variables — Extends independence to n random variables (joint PMF factors into the product of marginals), introduces i.i.d. random variables, and applies these ideas to the memoryless property of the geometric distribution.
- Functions of One Random Variable: One-to-One Transformations — For one-to-one functions of a single discrete random variable, the table method shows that probabilities are simply relabeled without change, and stem plots help visualize how the PMF shifts or stretches.
- Functions of One Random Variable: Many-to-One Transformations — When a function maps multiple values of a discrete random variable to the same output, the PMF of the resulting variable is found by summing probabilities over all inputs sharing each output value.
- Functions of Two Random Variables: Contours, Sum, and Max — The PMF of a function of two discrete random variables is found by identifying its range and summing the joint PMF over all (X, Y) pairs lying on each contour; visualizing contours for sum, max, and min helps organize these computations.
- Convolution, Key Distribution Relationships, and Min/Max of Random Variables — The PMF of the sum of independent integer-valued random variables is given by convolution; important special cases include Bernoulli sums yielding Binomial, Poisson sums yielding Poisson, and the CDF factorization law for the max/min of independent random variables.
Week 3
- Expected Value of Discrete Random Variables — Defines expected value (mean) of a discrete random variable as the probability-weighted sum of its range values, and computes it for Bernoulli, uniform, geometric, Poisson, and binomial distributions using summation tools.
- Properties of Expected Value — Establishes key properties of expected value including the Law of the Unconscious Statistician (LOTUS), linearity, translation/centering, and applies them to binomial distributions and indicator random variables.
- Simulation of Expected Value — Demonstrates through Python simulations that the long-run average of repeated observations of a random variable converges to its theoretical expected value, using casino dice games, common distributions, and the balls-and-bins problem.
- Variance and Standard Deviation — Defines variance and standard deviation as measures of spread, derives their properties (scaling, translation invariance, additivity under independence), computes them for common distributions, and introduces standardized random variables.
- Covariance and Correlation — Introduces covariance and the correlation coefficient as measures of the linear relationship between two random variables, establishes their properties, and distinguishes independence from being uncorrelated.
- Markov and Chebyshev Inequalities — Uses mean and variance to derive Markov's inequality (for non-negative random variables) and Chebyshev's inequality, providing precise probability bounds on how far a random variable deviates from its mean.
Week 4
- Motivation for Continuous Random Variables — Explains why continuous random variables are needed when discrete alphabets grow large, using meteorite weights and the binomial distribution as motivating examples, and introduces the key idea of moving from individual values to intervals (histograms).
- Cumulative Distribution Function — Defines the CDF, establishes its key properties (non-decreasing, starts at 0, ends at 1), shows how to compute it for discrete random variables (Bernoulli, die, uniform 1–100), and demonstrates how to calculate probabilities of intervals using the CDF.
- Approximating Discrete CDF with a Continuous Function — Shows how the CDF of discrete distributions (binomial, meteorite data) approaches a smooth continuous curve as the alphabet grows, motivates replacing the stepwise CDF with a continuous function, and demonstrates simpler interval-probability calculations using a continuous CDF approximation.
- General and Continuous Random Variables — Defines general random variables via their CDF, explains when the probability at a single point is zero versus positive (jumps vs. continuity), formally defines continuous random variables as those with a everywhere-continuous CDF, and classifies CDFs as discrete, continuous, or mixed.
- Probability Density Function — Defines the PDF as the derivative of the CDF, establishes its properties (non-negative, integrates to 1, piecewise continuous), shows how to compute probabilities of events by integrating the PDF, and works through examples including finding the normalizing constant.
- Common Continuous Distributions — Introduces the uniform, exponential (including the memoryless property), and normal (Gaussian) distributions with their PDFs, CDFs, and probability calculations, and covers standardization of the normal distribution for table-based computation.
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