Independent random variables - StatLectTwo random variables are independent if they convey no information about each
other and, as a consequence, receiving information about one of the two does ...

Probability mass function - StatlectThe distribution of a discrete random variable can be characterized through its
probability mass function ( pmf ). The probability that a given number will be the ...

Fundamentals of statistics - StatLectFundamentals of statistics . Learn the mathematical foundations of statistics,
through a series of rigorous but accessible lectures on the most frequently
utilized ...

In formal terms, the probability mass function of a discrete random variable X ...
Example . Suppose a random variable X can take only three values (1, 2 and 3), ...

Expected value - StatlectExpected value . The concept of expected value of a random variable is one of the
most important concepts in probability theory. It was first devised in the 17th ...

Maximum likelihood - StatlectThis lecture deals with an estimation method called maximum likelihood (or
maximum likelihood estimation - MLE). Before reading this lecture you should be
...

Fundamentals of probability - StatlectThis is an introduction to the main concepts of probability theory. After reading the
lectures, test your knowledge with the multiple choice tests in the last section.

Joint moment generating function - StatlectAs an example, we derive the joint mgf of a standard multivariate normal random
vector . Example Let X be a Kx1 standard multivariate normal random vector .

Chi - square distribution values - StatlectHow to compute values of the chi - square distribution using tables or computer
programs such as Excel and Matlab.

This lecture deals with an estimation method called maximum likelihood (or
maximum ... that we use to make statements about the probability distribution that
...

Probability density function - StatlectFor example , in the picture below the blue line is the pdf of a normal random ...
Example . Suppose a random variable X has probability density function [eq5].

Chi -square distribution plots - Statlectthe first graph (red line ) is the probability density function of a Chi -square ... Plots
the grid for x for j=xtick line ('xData',[j j],'yData',ylimit,' color ',[0.75 0.75 0.75] ...

How Wikipedia could make the world an even better place - StatlectThis short post explains how websites like Wikipedia could help solve important
computational problems by harnessing the computational power of their ...

Combinations - StatlectCombinations with and without repetition, Definition and intuitive explanation. ...
The following is a useful recursive formula for computing binomial coefficients: ...

Gamma function - Exercise Set 1 - StatlectThis exercise set contains some solved exercises on the Gamma function . The
theory needed to solve these exercises is introduced in the lecture entitled ...

Mean square convergence - StatLectis said to converge to X in mean-square if [eq1] converges to X according to the
metric [eq4] defined as follows: [eq5] (if you do not understand what it means "to ...

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

Central Limit Theorem for correlated sequences - Proof - StatlectThis page sketches some ideas for a proof of a Central Limit Theorem for
correlated sequences. This is preliminary and exploratory. Let [eq1] be a
sequence of ...

Central Limit Theorem - Exercise Set 1 - StatLectThis exercise set contains some solved exercises on the Central Limit Theorem.
The theory needed to solve these exercises is introduced in the lecture entitled ...

Hypothesis testing - StatLectTest of hypothesis , null hypothesis , alternative hypothesis , critical region, test
statistic, power function, size of a test .

Precision matrix - StatlectGlossary entry for the term: precision matrix . ... tends to infinity , we have zero
precision; on the contrary, when variance tends to zero, we have infinite precision
. ... because, by elementary properties of the determinant , we have that [eq6].

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

Legitimate probability density functions - StatlectLegitimate probability density functions. This lecture discusses two properties
characterizing probability density functions (pdfs). Not only any pdf satisfies these
...

Conditional probability as a random variable - StatlectSix numbers (from 1 to $6)$ .... More details. Conditioning with respect to sigma -
algebras ... smallest $ sigma $ - algebra containing all the events $Hin ciFourier $
...

Probability - StatlectDefinitions and explanations of event, sample space , outcome, realized outcome,
probability , probability measure, probability space, properties of probability .

Central Limit Theorem - StatlectA Central Limit Theorem (CLT) is a proposition stating a set of conditions that are
sufficient to guarantee the convergence of the sample mean Xbar_n ...

Limit of a sequence - StatlectConvergence of a sequence of real numbers . Convergence of a generic
sequence of objects: definition and intuitive explanation. Subsequences, metrics
...

How to compute values of the normal distribution using tables or computer
programs such as ... A fact that is often used in calculations is the following: [eq5].

Covariance - StatlectCovariance : definition, intuition, computation, proofs, exercises. ... The following
covariance formula is often used to compute the covariance between two ...

Sums of independent random variables - StatLectWe explain first how to derive the distribution function of the sum and then how to
derive its probability mass function (if the summands are discrete ) or its ...