Your random variable, X could be equal to 1 if you get a six and 0 if you get any other number. This is just an exampleyou can define X and Y however you like (i.e. 2 if you roll a six and 9 if you don’t). A few more example of random variables: X = total of lotto numbers. The definition of the distribution Function of a random variable and probability Density continuous random variable. These concepts are widely used in articles about the site's statistics www.excel-in-practice.com. Examples of calculation of distribution Function and probability Density with the help of the MS EXCEL.
Two Types of Random Variables
A random variable [latex]text{x}[/latex], and its distribution, can be discrete or continuous.
Learning Objectives
Contrast discrete and continuous variables
Key Takeaways
Key Points
- A random variable is a variable taking on numerical values determined by the outcome of a random phenomenon.
- The probability distribution of a random variable [latex]text{x}[/latex] tells us what the possible values of [latex]text{x}[/latex] are and what probabilities are assigned to those values.
- A discrete random variable has a countable number of possible values.
- The probability of each value of a discrete random variable is between 0 and 1, and the sum of all the probabilities is equal to 1.
- A continuous random variable takes on all the values in some interval of numbers.
- A density curve describes the probability distribution of a continuous random variable, and the probability of a range of events is found by taking the area under the curve.
Key Terms
- random variable: a quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a die
- discrete random variable: obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, ….
- continuous random variable: obtained from data that can take infinitely many values
Random Variables
In probability and statistics, a randomvariable 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 the possible outcomes of a past experiment whose already-existing value is uncertain (for example, as a result of incomplete information or imprecise measurements). They may also conceptually represent either the results of an “objectively” random process (such as rolling a die), or the “subjective” randomness that results from incomplete knowledge of a quantity.
Random variables can be classified as either discrete (that is, taking any of a specified list of exact values) or as continuous (taking 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.
Discrete Random Variables
Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers). Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability. For example, the value of [latex]text{x}_1[/latex] takes on the probability [latex]text{p}_1[/latex], the value of [latex]text{x}_2[/latex] takes on the probability [latex]text{p}_2[/latex], and so on. The probabilities [latex]text{p}_text{i}[/latex] must satisfy two requirements: every probability [latex]text{p}_text{i}[/latex] is a number between 0 and 1, and the sum of all the probabilities is 1. ([latex]text{p}_1+text{p}_2+dots + text{p}_text{k} = 1[/latex])
Discrete Probability Disrtibution: This shows the probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.
Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of 100.
Continuous Random Variables
Continuous random variables, on the other hand, take on values that vary continuously within one or more real intervals, and have a cumulative distribution function (CDF) that is absolutely continuous. As a result, the random variable has an uncountable infinite number of possible values, all of which have probability 0, though ranges of such values can have nonzero probability. The resulting probability distribution of the random variable can be described by a probability density, where the probability is found by taking the area under the curve.
Probability Density Function: The image shows the probability density function (pdf) of the normal distribution, also called Gaussian or “bell curve”, the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values corresponds to the area under the curve.
Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities.
Probability Distributions for Discrete Random Variables
Probability distributions for discrete random variables can be displayed as a formula, in a table, or in a graph.
Learning Objectives
Give examples of discrete random variables
Key Takeaways
Key Points
- A discrete probability function must satisfy the following: [latex]0 leq text{f}(text{x}) leq 1[/latex], i.e., the values of [latex]text{f}(text{x})[/latex] are probabilities, hence between 0 and 1.
- A discrete probability function must also satisfy the following: [latex]sum text{f}(text{x}) = 1[/latex], i.e., adding the probabilities of all disjoint cases, we obtain the probability of the sample space, 1.
- The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable. The only difference is how it looks graphically.
Key Terms
- discrete random variable: obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, ….
- probability distribution: A function of a discrete random variable yielding the probability that the variable will have a given value.
- probability mass function: a function that gives the relative probability that a discrete random variable is exactly equal to some value
A discrete random variable [latex]text{x}[/latex] has a countable number of possible values. The probability distribution of a discrete random variable [latex]text{x}[/latex] lists the values and their probabilities, where value [latex]text{x}_1[/latex] has probability [latex]text{p}_1[/latex], value [latex]text{x}_2[/latex] has probability [latex]text{x}_2[/latex], and so on. Every probability [latex]text{p}_text{i}[/latex] is a number between 0 and 1, and the sum of all the probabilities is equal to 1.
Examples of discrete random variables include:
- The number of eggs that a hen lays in a given day (it can’t be 2.3)
- The number of people going to a given soccer match
- The number of students that come to class on a given day
- The number of people in line at McDonald’s on a given day and time
A discrete probability distribution can be described by a table, by a formula, or by a graph. For example, suppose that [latex]text{x}[/latex] is a random variable that represents the number of people waiting at the line at a fast-food restaurant and it happens to only take the values 2, 3, or 5 with probabilities [latex]frac{2}{10}[/latex], [latex]frac{3}{10}[/latex], and [latex]frac{5}{10}[/latex] respectively. This can be expressed through the function [latex]text{f}(text{x})= frac{text{x}}{10}[/latex], [latex]text{x}=2, 3, 5[/latex] or through the table below. Of the conditional probabilities of the event [latex]text{B}[/latex] given that [latex]text{A}_1[/latex] is the case or that [latex]text{A}_2[/latex] is the case, respectively. Notice that these two representations are equivalent, and that this can be represented graphically as in the probability histogram below.
Probability Histogram: This histogram displays the probabilities of each of the three discrete random variables.
The formula, table, and probability histogram satisfy the following necessary conditions of discrete probability distributions:
- [latex]0 leq text{f}(text{x}) leq 1[/latex], i.e., the values of [latex]text{f}(text{x})[/latex] are probabilities, hence between 0 and 1.
- [latex]sum text{f}(text{x}) = 1[/latex], i.e., adding the probabilities of all disjoint cases, we obtain the probability of the sample space, 1.
Sometimes, the discrete probability distribution is referred to as the probability mass function (pmf). The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable. The only difference is how it looks graphically.
Probability Mass Function: This shows the graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.
Discrete Probability Distribution: This table shows the values of the discrete random variable can take on and their corresponding probabilities.
Expected Values of Discrete Random Variables
The expected value of a random variable is the weighted average of all possible values that this random variable can take on.
Learning Objectives
Calculate the expected value of a discrete random variable
Key Takeaways
Key Points
- The expected value of a random variable [latex]text{X}[/latex] is defined as: [latex]text{E}[text{X}] = text{x}_1text{p}_1 + text{x}_2text{p}_2 + dots + text{x}_text{i}text{p}_text{i}[/latex], which can also be written as: [latex]text{E}[text{X}] = sum text{x}_text{i}text{p}_text{i}[/latex].
- If all outcomes [latex]text{x}_text{i}[/latex] are equally likely (that is, [latex]text{p}_1=text{p}_2=dots = text{p}_text{i}[/latex]), then the weighted average turns into the simple average.
- The expected value of [latex]text{X}[/latex] is what one expects to happen on average, even though sometimes it results in a number that is impossible (such as 2.5 children).
Key Terms
- discrete random variable: obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, ….
- expected value: of a discrete random variable, the sum of the probability of each possible outcome of the experiment multiplied by the value itself
Discrete Random Variable
A discrete random variable [latex]text{X}[/latex] has a countable number of possible values. The probability distribution of a discrete random variable [latex]text{X}[/latex] lists the values and their probabilities, such that [latex]text{x}_text{i}[/latex] has a probability of [latex]text{p}_text{i}[/latex]. The probabilities [latex]text{p}_text{i}[/latex] must satisfy two requirements:
- Every probability [latex]text{p}_text{i}[/latex] is a number between 0 and 1.
- The sum of the probabilities is 1: [latex]text{p}_1+text{p}_2+dots + text{p}_text{i} = 1[/latex].
Expected Value Definition
In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on. The weights used in computing this average are probabilities in the case of a discrete random variable.
The expected value may be intuitively understood by the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity. More informally, it can be interpreted as the long-run average of the results of many independent repetitions of an experiment (e.g. a dice roll). The value may not be expected in the ordinary sense—the “expected value” itself may be unlikely or even impossible (such as having 2.5 children), as is also the case with the sample mean.
How To Calculate Expected Value
Suppose random variable [latex]text{X}[/latex] can take value [latex]text{x}_1[/latex] with probability [latex]text{p}_1[/latex], value [latex]text{x}_2[/latex] with probability [latex]text{p}_2[/latex], and so on, up to value [latex]text{x}_i[/latex] with probability [latex]text{p}_i[/latex]. Then the expectation value of a random variable [latex]text{X}[/latex] is defined as: [latex]text{E}[text{X}] = text{x}_1text{p}_1 + text{x}_2text{p}_2 + dots + text{x}_text{i}text{p}_text{i}[/latex], which can also be written as: [latex]text{E}[text{X}] = sum text{x}_text{i}text{p}_text{i}[/latex].
If all outcomes [latex]text{x}_text{i}[/latex] are equally likely (that is, [latex]text{p}_1 = text{p}_2 = dots = text{p}_text{i}[/latex]), then the weighted average turns into the simple average. This is intuitive: the expected value of a random variable is the average of all values it can take; thus the expected value is what one expects to happen on average. If the outcomes [latex]text{x}_text{i}[/latex] are not equally probable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than the others. The intuition, however, remains the same: the expected value of [latex]text{X}[/latex] is what one expects to happen on average.
For example, let [latex]text{X}[/latex] represent the outcome of a roll of a six-sided die. The possible values for [latex]text{X}[/latex] are 1, 2, 3, 4, 5, and 6, all equally likely (each having the probability of [latex]frac{1}{6}[/latex]). The expectation of [latex]text{X}[/latex] is: [latex]text{E}[text{X}] = frac{1text{x}_1}{6} + frac{2text{x}_2}{6} + frac{3text{x}_3}{6} + frac{4text{x}_4}{6} + frac{5text{x}_5}{6} + frac{6text{x}_6}{6} = 3.5[/latex]. In this case, since all outcomes are equally likely, we could have simply averaged the numbers together: [latex]frac{1+2+3+4+5+6}{6} = 3.5[/latex].
Average Dice Value Against Number of Rolls: An illustration of the convergence of sequence averages of rolls of a die to the expected value of 3.5 as the number of rolls (trials) grows.
Part of a series on statistics |
Probability theory |
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In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon.[1] The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers.[2]
This graph shows how random variable is a function from all possible outcomes to numerical quantities and also how it is used for defining probability mass functions.
A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of imprecise measurements or quantum uncertainty). They may also conceptually represent either the results of an 'objectively' random process (such as 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 is instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.
As a function, a random variable is required to be measurable, which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions. Which outcome will be observed is not certain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
The domain of a random variable is a sample space, which is interpreted as the set of possible outcomes of a random phenomenon. For example, in the case of a coin toss, only two possible outcomes are considered, namely heads or tails.
A random variable has a probability distribution, which specifies the probability of its values. Random variables can be discrete, that is, taking any of a specified finite or countable list of values, endowed with a probability mass function characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both types.
Two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.
- 1Definition
- 3Examples
- 3.1Discrete random variable
- 4Measure-theoretic definition
- 6Functions of random variables
- 7Equivalence of random variables
- 11References
Definition[edit]
A random variable is a measurable function from a set of possible outcomes to a measurable space. The technical axiomatic definition requires to be a sample space of a probability triple (see the measure-theoretic definition).
The probability that takes on a value in a measurable set is written as
- ,
where is the probability measure on .
Standard case[edit]
In many cases, is real-valued, i.e. . In some contexts, the term random element (see extensions) is used to denote a random variable not of this form.
When the image (or range) of is countable, the random variable is called a discrete random variable[3]:399 and its distribution can be described by a probability mass function that assigns a probability to each value in the image of . If the image is uncountably infinite then is called a continuous random variable. In the special case that it is absolutely continuous, its distribution can be described by a probability density function, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous,[4] for example a mixture distribution. Such random variables cannot be described by a probability density or a probability mass function.
Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value.
Extensions[edit]
The term 'random variable' in statistics is traditionally limited to the real-valued case (). In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.
However, the definition above is valid for any measurable space of values. Thus one can consider random elements of other sets , such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions. One may then specifically refer to a random variable of type, or an -valued random variable.
This more general concept of a random element is particularly useful in disciplines such as graph theory, machine learning, natural language processing, and other fields in discrete mathematics and computer science, where one is often interested in modeling the random variation of non-numerical data structures. In some cases, it is nonetheless convenient to represent each element of using one or more real numbers. In this case, a random element may optionally be represented as a vector of real-valued random variables (all defined on the same underlying probability space , which allows the different random variables to covary). For example:
- A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector whose length equals the size of the vocabulary, where the only values of positive probability are , , and the position of the 1 indicates the word.
- A random sentence of given length may be represented as a vector of random words.
- A random graph on given vertices may be represented as a matrix of random variables, whose values specify the adjacency matrix of the random graph.
- A random function may be represented as a collection of random variables , giving the function's values at the various points in the function's domain. The are ordinary real-valued random variables provided that the function is real-valued. For example, a stochastic process is a random function of time, a random vector is a random function of some index set such as , and random field is a random function on any set (typically time, space, or a discrete set).
Distribution functions[edit]
If a random variable defined on the probability space is given, we can ask questions like 'How likely is it that the value of is equal to 2?'. This is the same as the probability of the event which is often written as or for short.
Recording all these probabilities of output ranges of a real-valued random variable yields the probability distribution of . The probability distribution 'forgets' about the particular probability space used to define and only records the probabilities of various values of . Such a probability distribution can always be captured by its cumulative distribution function
and sometimes also using a probability density function, . In measure-theoretic terms, we use the random variable to 'push-forward' the measure on to a measure on .The underlying probability space 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 or independence based on a joint distribution of two or more random variables on the same probability space. In practice, one often disposes of the space altogether and just puts a measure on that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on quantile functions for fuller development.
Examples[edit]
Discrete random variable[edit]
In an experiment a person may be chosen at random, and one random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to the person's height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.
Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be 'an even number of children'. For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum .
In examples such as these, the sample space is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.
If are countable sets of real numbers, and , then is a discrete distribution function. Here for , for . Taking for instance an enumeration of all rational numbers as , one gets a discrete distribution function that is not a step function or piecewise constant.[3]
Coin toss[edit]
The possible outcomes for one coin toss can be described by the sample space . We can introduce a real-valued random variable that models a $1 payoff for a successful bet on heads as follows:
If the coin is a fair coin, Y has a probability mass function given by:
Dice roll[edit]
If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sum S of the numbers on the two dice, then S is a discrete random variable whose distribution is described by the probability mass function plotted as the height of picture columns here.
A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers n1 and n2 from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable X given by the function that maps the pair to the sum:
and (if the dice are fair) has a probability mass function ƒX given by:
Continuous random variable[edit]
Formally, a continuous random variable is a random variable whose cumulative distribution function is continuous everywhere.[5] There are no 'gaps', which would correspond to numbers which have a finite probability of occurring. Instead, continuous random variables almost never take an exact prescribed value c (formally, ) but there is a positive probability that its value will lie in particular intervals which can be arbitrarily small. Continuous random variables usually admit probability density functions (PDF), which characterize their CDF and probability measures; such distributions are also called absolutely continuous; but some continuous distributions are singular, or mixes of an absolutely continuous part and a singular part.
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, X = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any range of values. For example, the probability of choosing a number in [0, 180] is 1⁄2. Instead of speaking of a probability mass function, we say that the probability density of X 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.
Given any interval[nb 1], a random variable called a 'continuous uniform random variable' (CURV) is defined to take any value in the interval with equal likelihood.[nb 2] The probability of falling in any subinterval [nb 1] is proportional to the length of the subinterval, specifically
where the denominator comes from the unitarity axiom of probability. The probability density function of a CURV is given by the indicator function of its interval of support normalized by the interval's length:
Of particular interest is the uniform distribution on the unit interval. Samples of any desired probability distribution can be generated by calculating the quantile function of on a randomly-generated number distributed uniformly on the unit interval. This exploits properties of cumulative distribution functions, which are a unifying framework for all random variables.Mixed type[edit]
A mixed random variable is a random variable whose cumulative distribution function is neither piecewise-constant (a discrete random variable) nor everywhere-continuous.[5] It can be realized as the sum of a discrete random variable and a continuous random variable; in which case the CDF will be the weighted average of the CDFs of the component variables.[5]
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, X = −1; otherwise X = the value of the spinner as in the preceding example. There is a probability of 1⁄2 that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example.
Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see Lebesgue's decomposition theorem § Refinement. The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
Measure-theoretic definition[edit]
The most formal, axiomatic definition of a random variable 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.[2]
The measure-theoretic definition is as follows.
Let be a probability space and a measurable space. Then an -valued random variable is a measurable function , which means that, for every subset , its preimage where .[6] This definition enables us to measure any subset in the target space by looking at its preimage, which by assumption is measurable.
In more intuitive terms, a member of is a possible outcome, a member of is a measurable subset of possible outcomes, the function gives the probability of each such measurable subset, represents the set of values that the random variable can take (such as the set of real numbers), and a member of is a 'well-behaved' (measurable) subset of (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.
When is a topological space, then the most common choice for the σ-algebra is the Borel σ-algebra, which is the σ-algebra generated by the collection of all open sets in . In such case the -valued random variable is called the -valued random variable. Moreover, when space is the real line , then such a real-valued random variable is called simply the random variable.
Real-valued random variables[edit]
In this case the observation space is the set of 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 σ-algebra on the set of 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 .
Moments[edit]
The 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 , and also called the first moment. In general, is not equal to . Once the 'average value' is known, one could then ask how far from this average value the values of typically are, a question that is answered by the variance and standard deviation of a random variable. can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of .
Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables , find a collection of functions such that the expectation values fully characterise the distribution of the random variable .
Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). 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 has the value 'green', 0 otherwise. Then, the expected value and other moments of this function can be determined.
Functions of random variables[edit]
A new random variable Y can be defined by applying a real Borel measurable function to the outcomes of a real-valued random variable . That is, . The cumulative distribution function of is then
If function is invertible (i.e., exists, where is 's inverse function) and is either increasing or decreasing, then the previous relation can be extended to obtain
With the same hypotheses of invertibility of , assuming also differentiability, the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to , in order to obtain[5]
If there is no invertibility of but each admits at most a countable number of roots (i.e., a finite, or countably infinite, number of such that ) then the previous relation between the probability density functions can be generalized with
where , according to the inverse function theorem. The formulas for densities do not demand to be increasing.
In the measure-theoretic, axiomatic approach to probability, if a random variable on and a Borel measurable function, then is also a random variable on , since the composition of measurable functions is also measurable. (However, this is not necessarily true if is Lebesgue measurable.[citation needed]) The same procedure that allowed one to go from a probability space to can be used to obtain the distribution of .
Example 1[edit]
Let be a real-valued, continuous random variable and let .
If , then , so
If , then
so
Example 2[edit]
Suppose 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
which is the cumulative distribution function (CDF) of an exponential distribution.
Example 3[edit]
Suppose 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.
Example 4[edit]
Suppose is a random variable with a 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). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:
The inverse transformation is
and its derivative is
Then,
This is a noncentral chi-squared distribution with one degree of freedom.
Equivalence of random variables[edit]
There 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 distribution[edit]
If the sample space is a subset of the real line, random variables X and Y are equal in distribution (denoted ) if they have the same distribution functions:
To be equal in distribution, random variables need not be defined on the same probability space. 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 independent, identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform.
Almost sure equality[edit]
Two random variables X and Y are equal almost surely (denoted ) 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[edit]
Finally, the two random variables X and Y are equal if they are equal as functions on their measurable space:
This notion is typically the least useful in probability theory because in practice and in theory, the underlying measure space of the experiment is rarely explicitly characterized or even characterizable.
Convergence[edit]
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 of random variables can converge to a random variable . These are explained in the article on convergence of random variables.
Notes[edit]
- ^ abThe interval I can be closed (of the form ), open () or clopen (of the form or ). The singleton sets and have measure zero and so are equivalent from the perspective of the Lebesgue measure and measures absolutely continuous with respect to it.
- ^Formally, given any subsets of equal Lebesgue measure, the probabilities that X is contained in and are equal: .
See also[edit]
- Random number generator produces a random value
References[edit]
- ^Blitzstein, Joe; Hwang, Jessica (2014). Introduction to Probability. CRC Press. ISBN9781466575592.
- ^ abSteigerwald, Douglas G. 'Economics 245A – Introduction to Measure Theory'(PDF). University of California, Santa Barbara. Retrieved April 26, 2013.
- ^ abYates, Daniel S.; Moore, David S; Starnes, Daren S. (2003). The Practice of Statistics (2nd ed.). New York: Freeman. ISBN978-0-7167-4773-4. Archived from the original on 2005-02-09.Cite uses deprecated parameter
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(help) - ^L. Castañeda; V. Arunachalam & S. Dharmaraja (2012). Introduction to Probability and Stochastic Processes with Applications. Wiley. p. 67.
- ^ abcdBertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN188652940X. OCLC51441829.
- ^Fristedt & Gray (1996, page 11)
Literature[edit]
- Fristedt, Bert; Gray, Lawrence (1996). A modern approach to probability theory. Boston: Birkhäuser. ISBN3-7643-3807-5.
- Kallenberg, Olav (1986). Random Measures (4th ed.). Berlin: Akademie Verlag. ISBN0-12-394960-2. MR0854102.
- Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Berlin: Springer Verlag. ISBN0-387-95313-2.
- Papoulis, Athanasios (1965). Probability, Random Variables, and Stochastic Processes (9th ed.). Tokyo: McGraw–Hill. ISBN0-07-119981-0.
External links[edit]
- Hazewinkel, Michiel, ed. (2001) [1994], 'Random variable', Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN978-1-55608-010-4
- Zukerman, Moshe (2014), Introduction to Queueing Theory and Stochastic Teletraffic Models(PDF)
- Zukerman, Moshe (2014), Basic Probability Topics(PDF)
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