Digital textbook on probability and statistics

Statlect is a free digital textbook on probability theory and mathematical statistics. Explore its main sections.

Fundamentals of probability theory Read a rigorous yet accessible introduction to the main concepts of probability theory, such as random variables, expected value, variance, correlation, conditional probability.

Probability distributions Explore this compendium of common probability distributions, including the binomial, Poisson, uniform, exponential and normal distributions.

Asymptotic theory Learn about stochastic convergence, including convergence in probability, almost surely and in distribution; read about the Central Limit Theorem and the Law of Large Numbers.

Fundamentals of statistics This is a rigorous introduction to the basics of mathematical statistics; learn about statistical inference, point estimation, interval estimation and hypothesis testing.

Glossary of probability and statistics terms Use this glossary to review the most important technical terms that are introduced in the digital textbook. Some glossary entries also contain additional explanations and examples.

Mathematical tools

Learn about mathematical concepts that are frequently used in probability theory and statistics.

Matrix algebra This is a collection of lectures on the most important topics in matrix algebra: matrix addition and multiplication; linear combinations; linear independence, rank and span; linear systems.

Other mathematical tools Review the basics of calculus, learn about the fundamentals of combinatorial analysis, such as permutations and combinations; discover special functions used in statistics.

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Maximum likelihood Maximum likelihood is an estimation method that allows to use observed data to estimate the parameters of the probability distribution that generated the data.

Exponential distribution The exponential distribution is a continuous probability distribution used to model the time we need to wait before a given event occurs.

Beta distribution The Beta distribution is a continuous probability distribution having two parameters. One of its most common uses is to model one's uncertainty about the probability of success of an experiment.

Poisson distribution The Poisson distribution is a discrete probability distribution used to model the number of occurrences of an unpredictable event within a unit of time.

Binomial distribution A discrete distribution used to model the number of successes obtained by repeating several times an experiment that can have two outcomes, either success or failure.

Moment generating function The moment generating function is often used to characterize the probability distribution of a random variable. Its derivatives at zero are equal to the moments of the random variable.

Convergence in probability The concept of convergence in probability is based on the following intuition: two random variables are "close to each other" if there is a high probability that their difference will be very small.

Wald test A test that is often performed on parameters that have been estimated by maximum likelihood, based on a test statistics called the Wald statistic.

Beta function The Beta function is often employed in probability theory and statistics, for example, as a normalizing constant in the density functions of the F and Student's t distributions.

Likelihood ratio test A statistical test based on the comparison of two parameter estimates, a restricted one and an unrestricted one.

Central Limit Theorems A Central Limit Theorem provides a set of conditions that are sufficient for the sample mean to have a normal distribution asymptotically (as the sample size increases).

Expected value A gentle introduction to the concept of expected value, with an informal definition and more formal definitions based on the Stieltjes and Lebesgue integrals.

Bayes' rule Bayes' rule is a formula that allows to compute the conditional probability of a given event, after observing a second event whose conditional and unconditional probabilities were known in advance.

Linear regression models The linear regression model is a conditional model in which the output variable is linearly related to the input variables and to an error term.

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Ridge regression The ridge estimator of the coefficients of a linear regression is biased but can have lower mean squared error than the OLS estimator.

Model selection criteria Model selection criteria, such as the Akaike Information Criterion (AIC) are used to select the best model among a set of candidate statistical models.

Logit model The logit model is a classification model used to predict the realization of a binary variable on the basis of a set of regressors.