Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Linear Combinations: Moment-Generating Functions

In course

MATH-232: Probability and statistics (for IC)

Cours d'introduction à la théorie des probabilités et aux statistiques. ...

Description

This lecture covers the concept of moment-generating functions, linear combinations of random variables, and the normality of linear combinations. It also delves into the distribution of sums of independent random variables and the properties of MGFs. Important theorems and examples are provided to illustrate the concepts.

Instructors (2)

Raphaël Jean Berthier

Emmanuel Abbé

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Related concepts (32)

Normal distribution

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.

Bayesian probability

Bayesian probability (ˈbeɪziən or ˈbeɪʒən ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown.

Moment-generating function

In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables.

Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem.

Related lectures (50)

Continuous Random Variables

Explores continuous random variables, density functions, joint variables, independence, and conditional densities.

Continuous Random Variables

Covers continuous random variables, probability density functions, and distributions, with practical examples.

Probability and Statistics

Covers probability distributions, moments, and continuous random variables.