**Introduction to Random Variables**

Introduction to random variables and probability distribution functions.

## Introduction to Random Variables

## PDF File,View to Click Hear

Introduction to Random Variables

Random variable:

From Wikipedia, the free encyclopedia

In probability and statistics, a

A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment or an event that has not happened yet, or the potential values of a past experiment or event whose already-existing value is uncertain (e.g. as a result of incomplete information or imprecise measurements). They may also conceptually represent either the results of an "objectively" random process (e.g. rolling a die), or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.

Random variables can be classified as either

The basic concept of "random variable" in statistics is real-valued. However, one can consider arbitrary types such as boolean values, categorical variables, complex numbers, vectors, matrices,sequences, trees, sets, shapes, manifolds, functions, and processes. The term

The formal mathematical treatment of random variables is dealt with in the subject of probability theory. In that context, random variables are defined in terms of functions defined on a probability space.

From Wikipedia, the free encyclopedia

In probability and statistics, a

**random variable**or**stochastic variable**is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense). As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment or an event that has not happened yet, or the potential values of a past experiment or event whose already-existing value is uncertain (e.g. as a result of incomplete information or imprecise measurements). They may also conceptually represent either the results of an "objectively" random process (e.g. rolling a die), or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.

Random variables can be classified as either

*discrete*(i.e. it may assume any of a specified list of exact values) or as*continuous*(i.e. it may assume any numerical value in an interval or collection of intervals). The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution. The realizations of a random variable, i.e. the results of randomly choosing values according to the variable's probability distribution are called random variates.The basic concept of "random variable" in statistics is real-valued. However, one can consider arbitrary types such as boolean values, categorical variables, complex numbers, vectors, matrices,sequences, trees, sets, shapes, manifolds, functions, and processes. The term

*random element*is used to encompass all such related concepts. An example is the stochastic process, a set of indexed random variables (typically indexed by time or space). These more general concepts are particularly useful in fields such as computer science and natural language processing where many of the basic elements of analysis are non-numerical. Such general random elements can sometimes be treated as sets of real-valued random variables — often more specifically as random vectors). For example:- A "random word" may be parameterized by an integer-valued index into the vocabulary of possible words; or alternatively as an indicator vector, in which exactly one element is a 1 and the others are 0, with the 1 indexing a particular word into a vocabulary.
- A "random sentence" may be parameterized as a vector of random words.
- A random graph, for a graph with
*V*edges, may be parameterized as an*NxN*matrix, indicating the weight for each edge, or 0 for no edge. (If the graph has no weights, 1 indicates an edge, 0 indicates no edge.)

The formal mathematical treatment of random variables is dealt with in the subject of probability theory. In that context, random variables are defined in terms of functions defined on a probability space.

Introduction:

Real-valued random variables (those whose range is the real numbers) are used in the sciences to make predictions based on data obtained from scientific experiments.[

Another example is the sample mean, which is the average of a number of samples. When these samples are independent observations of the same random event they can be called independent identically distributed random variables. Since each sample is a random variable, the sample mean is a function of random variables and hence a random variable itself, whose distribution can be computed and properties determined.

One of the reasons that real-valued random variables are so commonly considered is that the expected value (a type of average) and variance (a measure of the "spread", or extent to which the values are dispersed) of the variable can be computed.[

There are several types of random variables, the most common two are the discrete and the continuous.[1] A discrete random variable maps outcomes to values of a countable set (e.g., theintegers), with each value in the range having probability greater than or equal to zero. A continuous random variable maps outcomes to values of an uncountable set (e.g., the real numbers). For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values (such as an interval of non-zero length) may be positive. A random variable can be "mixed", with part of its probability spread out over an interval like a typical continuous variable, and part of it concentrated on particular values like a discrete variable. These classifications are equivalent to the categorization of probability distributions.

The expected value of random vectors, random matrices, and similar aggregates of fixed structure is defined as the aggregation of the expected value computed over each individual element. The concept of "variance of a random vector" is normally expressed through a covariance matrix. No generally-agreed-upon definition of expected value or variance exists for cases other than just discussed.

DefinitionA random variable is defined on a set of possible outcomes (the sample space Ω) and a probability distribution that associates each outcome with a probability. A random variable represents a measurable aspect or property of the outcomes, and hence associates each outcome with a number. In an experiment a person may be chosen at random, and one random variable may be its age, and another its number of children. Formally a random variable is considered to be a function on the possible outcomes. Random variables are typically classified as either discrete or continuous.Discrete variables can take on either a finite or at most a countably infinite set of discrete values. Their probability distribution is given by a probability mass function which directly maps a value of the random variable to a probability. Continuous variables, however, take on values that vary continuously within one or more (possibly infinite) intervals. As a result there are anuncountably infinite number of individual outcomes, and each has a probability 0. As a result, the probability distribution for many continuous random variables is defined using a probability density function, which indicates the "density" of probability in a small neighborhood around a given value. More technically, the probability that an outcome is in a particular range is derived from the integration of the probability density function in that range. Both concepts can be united using a cumulative distribution function (CDF), which describes the probability that an outcome will be less than or equal to a specified value. This function is necessarily monotonically non-decreasing, with a minimum value of 0 at negative infinity and a maximum value of 1 at positive infinity. The CDF of a discrete distribution will consist mostly of flat areas along with sudden jumps at each outcome defined in the sample space, while the CDF of a continuous distribution will typically rise gradually and continuously. Distributions that are partly discrete and partly continuous can also be described this way.

ExamplesIn a poll an adult person is chosen at random from the British population . The random variable

The possible outcomes for one coin toss can be described by the sample space = {heads, tails}. We can introduce a real-valued random variable

If the coin is equally likely to land on either side then it has a probability mass function given by:

A random variable can also be used to describe the process of rolling a die and the possible outcomes. The most obvious representation is to take the set = {1, 2, 3, 4, 5, 6} as the sample space, defining the random variable X to be equal to the number rolled. In this case,

and

An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case,

An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails,

Measure-theoretic definitionThe most formal, axiomatic definition of random variables involves measure theory. Continuous random variables are defined in terms of sets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the Banach-Tarski paradox) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a sigma-algebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the Borel σ-algebra, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals.

The measure-theoretic definition is as follows.

Let (Ω, ℱ,

When

Real-valued random variablesIn this case the observation space is the real numbers. Recall, is the probability space. For real observation space, the function is a real-valued random variable if

This definition is a special case of the above because the set generates the Borel sigma-algebra on the real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that .

Distribution functions of random variablesIf a random variable defined on the probability space is given, we can ask questions like "How likely is it that the value of is bigger than 2?". This is the same as the probability of the event which is often written as for short.

Recording all these probabilities of output ranges of a real-valued random variable

and sometimes also using a probability density function. In measure-theoretic terms, we use the random variable

MomentsThe probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted E[

Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables

Moments can only be defined for real-valued functions of random variables. If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable

Functions of random variablesA new random variable

If function

and, again with the same hypotheses of invertibility of

.If there is no invertibility of

where

In the measure-theoretic, axiomatic approach to probability, if we have a random variable on and a Borel measurable function , then will also be a random variable on , since the composition of measurable functions is also measurable. (However, this is not true if is Lebesgue measurable.) The same procedure that allowed one to go from a probability space to can be used to obtain the distribution of .

Example 1Let

If

If

so

Example 2Suppose is a random variable with a cumulative distribution

where is a fixed parameter. Consider the random variable Then,

The last expression can be calculated in terms of the cumulative distribution of so

Example 3Suppose is a random variable with a standard normal distribution, whose density is

Consider the random variable We can find the density using the above formula for a change of variables:

In this case the change is not monotonic, because every value of has two corresponding values of (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.

The inverse transformation is

and its derivative is

Then:

This is a chi-squared distribution with one degree of freedom.

Equivalence of random variablesThere are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.

In increasing order of strength, the precise definition of these notions of equivalence is given below.

Equality in distributionIf the sample space is a subset of the real line a possible definition is that random variables

Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of i.i.d. random variables. However, the moment generating function exists only for distributions that are good enough.[

Almost sure equalityTwo random variables

For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:

where "ess sup" represents the essential supremum in the sense of measure theory.

Equality:

Finally, the two random variables

Convergence:

A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; for instance the law of large numbers and the central limit theorem.

There are various senses in which a sequence (

See also Statistics portal

References:

^ Taken the references from Wikipedia 08 Sep,2012

^ Taken the references from Youtub.com , 2012

Real-valued random variables (those whose range is the real numbers) are used in the sciences to make predictions based on data obtained from scientific experiments.[

*citation needed*] In addition to scientific applications, random variables were developed for the analysis of games of chance and stochastic events. In such instances, the function that maps the outcome to a real number is often the identity function or similarly trivial function, and not explicitly described. In many cases, however, it is useful to consider random variables that are functions of other random variables, and then the mapping function included in the definition of a random variable becomes important. As an example, the square of a random variable distributed according to a standard normal distribution is itself a random variable, with a chi-squared distribution. One way to think of this is to imagine generating a large number of samples from a standard normal distribution, squaring each one, and plotting a histogram of the values observed. With enough samples, the graph of the histogram will approximate the density function of a chi-squared distribution with one degree of freedom.Another example is the sample mean, which is the average of a number of samples. When these samples are independent observations of the same random event they can be called independent identically distributed random variables. Since each sample is a random variable, the sample mean is a function of random variables and hence a random variable itself, whose distribution can be computed and properties determined.

One of the reasons that real-valued random variables are so commonly considered is that the expected value (a type of average) and variance (a measure of the "spread", or extent to which the values are dispersed) of the variable can be computed.[

*citation needed*]There are several types of random variables, the most common two are the discrete and the continuous.[1] A discrete random variable maps outcomes to values of a countable set (e.g., theintegers), with each value in the range having probability greater than or equal to zero. A continuous random variable maps outcomes to values of an uncountable set (e.g., the real numbers). For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values (such as an interval of non-zero length) may be positive. A random variable can be "mixed", with part of its probability spread out over an interval like a typical continuous variable, and part of it concentrated on particular values like a discrete variable. These classifications are equivalent to the categorization of probability distributions.

The expected value of random vectors, random matrices, and similar aggregates of fixed structure is defined as the aggregation of the expected value computed over each individual element. The concept of "variance of a random vector" is normally expressed through a covariance matrix. No generally-agreed-upon definition of expected value or variance exists for cases other than just discussed.

DefinitionA random variable is defined on a set of possible outcomes (the sample space Ω) and a probability distribution that associates each outcome with a probability. A random variable represents a measurable aspect or property of the outcomes, and hence associates each outcome with a number. In an experiment a person may be chosen at random, and one random variable may be its age, and another its number of children. Formally a random variable is considered to be a function on the possible outcomes. Random variables are typically classified as either discrete or continuous.Discrete variables can take on either a finite or at most a countably infinite set of discrete values. Their probability distribution is given by a probability mass function which directly maps a value of the random variable to a probability. Continuous variables, however, take on values that vary continuously within one or more (possibly infinite) intervals. As a result there are anuncountably infinite number of individual outcomes, and each has a probability 0. As a result, the probability distribution for many continuous random variables is defined using a probability density function, which indicates the "density" of probability in a small neighborhood around a given value. More technically, the probability that an outcome is in a particular range is derived from the integration of the probability density function in that range. Both concepts can be united using a cumulative distribution function (CDF), which describes the probability that an outcome will be less than or equal to a specified value. This function is necessarily monotonically non-decreasing, with a minimum value of 0 at negative infinity and a maximum value of 1 at positive infinity. The CDF of a discrete distribution will consist mostly of flat areas along with sudden jumps at each outcome defined in the sample space, while the CDF of a continuous distribution will typically rise gradually and continuously. Distributions that are partly discrete and partly continuous can also be described this way.

ExamplesIn a poll an adult person is chosen at random from the British population . The random variable

*A*is the age of the chosen person, and the random variable*C*the number of children:The possible outcomes for one coin toss can be described by the sample space = {heads, tails}. We can introduce a real-valued random variable

*Y*that models a $1 payoff for a successful bet on heads as follows:If the coin is equally likely to land on either side then it has a probability mass function given by:

A random variable can also be used to describe the process of rolling a die and the possible outcomes. The most obvious representation is to take the set = {1, 2, 3, 4, 5, 6} as the sample space, defining the random variable X to be equal to the number rolled. In this case,

and

An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case,

*= the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any***X***range*of values. For example, the probability of choosing a number in [0, 180] is ½. Instead of speaking of a probability mass function, we say that the probability*density*of*is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.***X**An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails,

*= −1; otherwise***X***= the value of the spinner as in the preceding example. There is a probability of ½ that this random variable will have the value −1. Other ranges of values would have half the probability of the last example.***X**Measure-theoretic definitionThe most formal, axiomatic definition of random variables involves measure theory. Continuous random variables are defined in terms of sets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the Banach-Tarski paradox) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a sigma-algebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the Borel σ-algebra, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals.

The measure-theoretic definition is as follows.

Let (Ω, ℱ,

*P*) be a probability space and (*E*, ℰ) a measurable space. Then an**(**is a function*E*, ℰ)-valued random variable*X*: Ω→*E*which is (ℱ, ℰ)-measurable. The latter means that, for every subset*B*∈ ℰ, its preimage*X*−1(*B*) ∈ ℱ where*X*−1(*B*) = {*ω*:*X*(*ω*) ∈*B*}.[2] This definition enables us to measure any subset*B*in the target space by looking at its preimage, which by assumption is measurable.When

*E*is a topological space, then the most common choice for the σ-algebra ℰ is to take it equal to the Borel σ-algebra ℬ(*E*), which is the σ-algebra generated by the collection of all open sets in*E*. In such case the (*E*, ℰ)-valued random variable is called the**. Moreover, when space***E*-valued random variable*E*is the real line ℝ, then such real-valued random variable is called simply the**random variable**.Real-valued random variablesIn this case the observation space is the real numbers. Recall, is the probability space. For real observation space, the function is a real-valued random variable if

This definition is a special case of the above because the set generates the Borel sigma-algebra on the real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that .

Distribution functions of random variablesIf a random variable defined on the probability space is given, we can ask questions like "How likely is it that the value of is bigger than 2?". This is the same as the probability of the event which is often written as for short.

Recording all these probabilities of output ranges of a real-valued random variable

*X*yields the probability distribution of*X*. The probability distribution "forgets" about the particular probability space used to define*X*and only records the probabilities of various values of*X*. Such a probability distribution can always be captured by its cumulative distribution functionand sometimes also using a probability density function. In measure-theoretic terms, we use the random variable

*X*to "push-forward" the measure*P*on Ω to a measure d*F*on**R**. The underlying probability space Ω is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence orindependence based on a joint distribution of two or more random variables on the same probability space. In practice, one often disposes of the space Ω altogether and just puts a measure on**R**that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables.MomentsThe probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted E[

*X*], and also called the**first moment.**In general, E[*f*(*X*)] is not equal to*f*(E[*X*]). Once the "average value" is known, one could then ask how far from this average value the values of*X*typically are, a question that is answered by the variance and standard deviation of a random variable. E[*X*] can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of*X*.Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables

*X*, find a collection {*fi*} of functions such that the expectation values E[*fi*(*X*)] fully characterise the distribution of the random variable*X*.Moments can only be defined for real-valued functions of random variables. If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable

*X*that can take on the nominal values "red", "blue" or "green", the real-valued function can be constructed; this uses the Iverson bracket, and has the value 1 if*X*has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined.Functions of random variablesA new random variable

*Y*can be defined by applying a real Borel measurable function to the outcomes of a real-valued random variable*X*. The cumulative distribution function of isIf function

*g*is invertible, i.e.*g*−1 exists, and increasing, then the previous relation can be extended to obtainand, again with the same hypotheses of invertibility of

*g*, assuming also differentiability, we can find the relation between the probability density functions by differentiating both sides with respect to*y*, in order to obtain.If there is no invertibility of

*g*but each*y*admits at most a countable number of roots (i.e. a finite, or countably infinite, number of*xi*such that*y = g(xi)*) then the previous relation between theprobability density functions can be generalized withwhere

*xi = gi-1(y)*. The formulas for densities do not demand*g*to be increasing.In the measure-theoretic, axiomatic approach to probability, if we have a random variable on and a Borel measurable function , then will also be a random variable on , since the composition of measurable functions is also measurable. (However, this is not true if is Lebesgue measurable.) The same procedure that allowed one to go from a probability space to can be used to obtain the distribution of .

Example 1Let

*X*be a real-valued, continuous random variable and let*Y*=*X*2.If

*y*< 0, then P(*X*2 ≤*y*) = 0, soIf

*y*≥ 0, thenso

Example 2Suppose is a random variable with a cumulative distribution

where is a fixed parameter. Consider the random variable Then,

The last expression can be calculated in terms of the cumulative distribution of so

Example 3Suppose is a random variable with a standard normal distribution, whose density is

Consider the random variable We can find the density using the above formula for a change of variables:

In this case the change is not monotonic, because every value of has two corresponding values of (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.

The inverse transformation is

and its derivative is

Then:

This is a chi-squared distribution with one degree of freedom.

Equivalence of random variablesThere are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.

In increasing order of strength, the precise definition of these notions of equivalence is given below.

Equality in distributionIf the sample space is a subset of the real line a possible definition is that random variables

*X*and*Y*are*equal in distribution*if they have the same distribution functions:Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of i.i.d. random variables. However, the moment generating function exists only for distributions that are good enough.[

*clarification needed*]Almost sure equalityTwo random variables

*X*and*Y*are*equal almost surely*if, and only if, the probability that they are different is zero:For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:

where "ess sup" represents the essential supremum in the sense of measure theory.

Equality:

Finally, the two random variables

*X*and*Y*are*equal*if they are equal as functions on their measurable space:Convergence:

A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; for instance the law of large numbers and the central limit theorem.

There are various senses in which a sequence (

*X**n*) of random variables can converge to a random variable*X*. These are explained in the article on convergence of random variables.See also Statistics portal

- Observable variable
- Probability distribution
- Algebra of random variables
- Multivariate random variable
- Event (probability theory)
- Randomness
- Random element
- Random vector
- Random function
- Random measure
- Stochastic process

References:

^ Taken the references from Wikipedia 08 Sep,2012

^ Taken the references from Youtub.com , 2012