Independent random variables - Statlect, the digital textbookTwo 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 - Statlect, the digital textbookFundamentals 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), ...

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 .

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

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

Probability - StatLectWe then discuss the properties that probability needs to satisfy. Finally, we
mention some possible interpretations of the concept of probability . Ω. In this
lecture ...

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 ...

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

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

Independent random variables - StatLectThis lecture provides a formal definition of independence and discusses how to
verify whether two or more random variables are independent .

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 ...

Laws of Large Numbers - Statlect, the digital textbookDefinition of Law of Large Numbers . Weak Law. Strong Law. Chebyshev's Weak
Law. Proofs. Exercises.

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

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

Normal distribution values - Statlect, the digital textbookHow to compute values of the normal distribution using tables or computer
programs such as Excel and Matlab.

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 ...

We then discuss the properties that probability needs to satisfy. Finally, we
mention some possible interpretations of the concept of probability. ... Six
numbers , from 1 to 6, can appear face up, but we do not yet know which one of
them will ...

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

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
...

Legitimate probability mass functions - StatLectWe prove not only that any probability mass function satisfies these two
properties, but also that any function satisfying these two properties is a
legitimate ...

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 ...

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

Point estimation - StatLectwhen the squared error is used as a loss function, then the risk [eq26] is called
mean squared error ( MSE ). The square root of the mean squared error is called ...

A 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 ...

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 ...

when the squared error is used as a loss function, then the risk [eq26] is called
mean squared error (MSE). The square root of the mean squared error is called ...

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 ...