Probability theory is the mathematical backbone of statistics and econometrics.
Much of this subject centres on random variables: quantities whose values are not yet known, but whose possible behaviour we can describe using probability. Understanding what these are – and what their distributions tell us – is essential for making sense of much of what comes later.
We begin with the basic rules of probability, before developing the main tools for studying random variables, their distributions, expectations, relationships, and limiting behaviour.
Some of these ideas can seem rather abstract at first. But getting them straight now will save us a great deal of confusion later!
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Probability Basics
The basic rules for calculating probabilities, including conditional probability and the law of total probability.
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Random Variables
What random variables are, how their distributions are expressed, and discrete and continuous cases.
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Expected Values and Variance
Key quantities that describe where a random variable tends to lie, and how widely its values are spread.
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Several Random Variables
Working with several random variables at once, including joint distributions, independence, covariance and correlation.
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Basic Distributions
The most important standard probability distributions, how to recognise them, and when each one arises.
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Conditioning
How learning about one random variable changes what we expect to happen with another.
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Advanced Distributions
The chi-squared, t- and F-distributions: where they come from and why they become important later.
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Generating Functions
A powerful way to package probabilities and moments into functions that are easier to manipulate.
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Convergence of Random Variables
Three different ways that random variables can settle down as a sample becomes large.
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Advanced Probability Concepts
A more careful look at discrete random variables, countability, and what a random variable really is.