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 ...

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), ...

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 ...

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
...

Poisson distribution - Statlect, the digital textbookThe Poisson distribution explained, with examples, solved exercises and detailed
proofs of important results.

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 ...

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

Normal distribution - Maximum likelihood estimation - StatlectMaximum likelihood estimation of the parameters of the normal distribution.
Derivation and properties, with detailed proofs.

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 .

Probability - StatLectSince we usually speak of the " probability of an event", the next section
introduces a formal definition of the concept of event. We then discuss the
properties that ...

Statlect, the digital textbook on probability and statisticsStatlect is a free digital textbook on probability theory and mathematical statistics .
Explore its main sections. Fundamentals of probability theory. Read a rigorous ...

Combinations - StatLectCombinations with and without repetition, Definition and intuitive explanation.
Counting combinations . Binomial coefficient. Examples .

Mean squared error - StatLectIn the theory of point estimation, the mean squared error is frequently used to
assess the risk of an estimator, that is, how large are on average the losses ...

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

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.

How Wikipedia could make the world an even better place - StatLectThis short post explains how websites like Wikipedia could help solve important
... It would suffice to add a piece of Javascript code to every Wikipedia page.

To keep things simple, we provide an informal definition of expected value and
we discuss its computation in this lecture, while we relegate a more rigorous ...

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 ...

This short post explains how websites like Wikipedia could help solve important
computational problems by harnessing the computational power of their ...

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 ...

Marginal probability density function - StatLectThis is called marginal probability density function , in order to distinguish it from
... Example . Let X be a $2 imes 1$ absolutely continuous random vector having ...

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] ...

Statlect is a free digital textbook on probability theory and mathematical statistics.
... Explore this compendium of common probability distributions , including the ...

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 ...

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

We then discuss the properties that probability needs to satisfy. ... Six numbers ,
from 1 to 6, can appear face up, but we do not yet know which one of them ....
clarify the meaning of probability, they all touch upon important aspects 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 ...

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].

Definitions and explanations of event, sample space , outcome, realized outcome,
probability , probability measure, probability space, properties of probability .

is large enough, then a standard normal distribution is a good approximation of
the ... So, roughly speaking, under the stated assumptions , the distribution of the
...

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 ...