Unit 4 of 12
Probability Models for Statistics
- 12 lessons
- Videos 17 to 28
Probability models describe chance, and statistical inference rests on them. This unit builds the probability later units use, from events and conditional probability to normal models and simulation.
The addition and multiplication rules, conditional probability and independence lead to Bayes' theorem, which shows how the base rate affects what a positive test result means. Counting rules handle sampling without replacement. Discrete random variables have expected values and variances, and the binomial, hypergeometric and Poisson distributions model counts. Continuous models assign probability by area under a density, such as the uniform, exponential and normal distributions. Normal approximations and their limits follow, and the unit ends with simulation and its Monte Carlo error.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Events, Conditioning and Counting
Probability Rules and Events
How a probability model assigns probabilities to events in a sample space, how the complement and addition rules work, what unions, intersections and disjoint events are, and how probability relates to relative frequency.
Video 17
Conditional Probability and Independence
How conditional probability restricts attention to a known event, how joint and marginal probabilities appear in a two-way table, and how the multiplication rule, independent events and the law of total probability build on them.
Video 18
Bayes' Theorem and Base Rates
How Bayes' theorem turns a prior probability into a posterior probability, why the base rate matters, and how it combines with sensitivity and specificity to give the positive predictive value, seen through natural frequencies.
Video 19
Counting and Sampling Without Replacement
How the multiplication principle, factorials, permutations and combinations count samples, when a ratio of counts is a probability, and how sampling without replacement differs from sampling with replacement.
Video 20
Discrete Random Variables and Models
Discrete Random Variables and Probability Distributions
What a random variable is, when it is discrete, and how its support, probability mass function and cumulative distribution function describe its probability distribution, including strict and inclusive events.
Video 21
Expected Value, Variance and Sums of Random Variables
How expected value and variance summarize a probability model, how the second moment and linearity of expectation simplify them, and how covariance and independence decide the variance of a sum.
Video 22
Bernoulli, Binomial and Hypergeometric Models
How a Bernoulli trial leads to the binomial distribution, which conditions a binomial setting needs, the binomial mean and variance, and when the hypergeometric distribution fits sampling a finite batch without replacement.
Video 23
Poisson Models for Event Counts
How the Poisson distribution models a count over a stated exposure, how an event rate and the exposure give the Poisson parameter, which process assumptions support the model, and how overdispersion shows a mismatch.
Video 24
Continuous and Normal Models
Continuous Distributions, Density and Cumulative Probability
How a continuous random variable assigns probability as area under a density, how to read a cumulative distribution function, and how the uniform and exponential models, survival probability and memorylessness work.
Video 25
Normal Distributions and Quantiles
How to standardize a value under a normal model, find tail probabilities with the standard normal CDF, reverse the calculation to get normal quantiles, and read the empirical rule as a model summary.
Video 26
Normal Approximations and Their Limits
How a normal curve with the binomial mean and variance approximates a count, how continuity correction matches integer events, why the success-failure condition is only a guideline, and how to measure approximation error.
Video 27
Simulation
Simulation and Monte Carlo Error
How a Monte Carlo simulation estimates a model probability from replications, how a random seed makes a run reproducible, and how Monte Carlo error and its standard error differ from model error.
Video 28