| Part of a series on statistics | |
| Probability theory | |
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| H or tails H mapped to −1 and X is a measurable function E. For the measurability of (Ω,ℱ,P) (see the measure-theoretic definition). A random variable is often denoted by capital Roman letters such as S⊆E is written as
E=ℝ. 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 X 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. 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. ExtensionseditThe term "random variable" in statistics is traditionally limited to the real-valued case (E of values. Thus one can consider random elements of other sets E-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 (1 0 0 0 ⋯), N may be represented as a vector of N×N matrix of random variables, whose values specify the adjacency matrix of the random graph. Distribution functionseditIf a random variable X is equal to 2?". This is the same as the probability of the event pX(2) for short. Recording all these probabilities of outputs of a random variable X and only records the probabilities of various output values of FX(x)=P(X≤x) and sometimes also using a probability density function, P on ℝ. The measure fX=dpX/dμ, the Radon–Nikodym derivative of ℝ (often, this reference measure is the Lebesgue measure in the case of continuous random variables, or the counting measure in the case of discrete random variables). The underlying probability space ℝ 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. ExampleseditDiscrete random variableeditConsider an experiment where a person is chosen at random. An example of a random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to their 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 {an},{bn} are countable sets of real numbers, 0"}}'>bn>0 and F=∑nbnδan(x) is a discrete distribution function. Here x<t, {an}, one gets a discrete function that is not necessarily a step function (piecewise constant). Coin tosseditThe possible outcomes for one coin toss can be described by the sample space Y(ω)={1,if ω=heads,0,if ω=tails. If the coin is a fair coin, fY given by: fX(S)=min(S−1,13−S)36, for S∈{2,3,4,5,6,7,8,9,10,11,12} Continuous random variableeditFormally, a continuous random variable is a random variable whose cumulative distribution function is continuous everywhere. 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, XI∼U(I)=U[a,b] is called a "continuous uniform random variable" (CURV) if the probability that it takes a value in a subinterval depends only on the length of the subinterval. This implies that the probability of Pr(XI∈[c,d])=d−cb−a where the last equality results from the unitarity axiom of probability. The probability density function of a CURV [0,1]. Samples of any desired probability distribution CDF will be the weighted average of the CDFs of the component variables. 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 is the value of the spinner as in the preceding example. There is a probability of (E,ℰ) a measurable space. Then an X:Ω→E, which means that, for every subset ℱ-measurable; B∈ℰ in the target space by looking at its preimage, which by assumption is measurable. In more intuitive terms, a member of P gives the probability of each such measurable subset, E (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 (E,ℰ)-valued random variable is called an ℝ, then such a real-valued random variable is called simply a random variable. Note that we are not giving σ-algebra. This choice allows for more measurable functions (Ω,ℱ,P) is the probability space. For a real observation space, the function {(−∞,r]:r∈ℝ} 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 E[X], and also called the first moment. In general, X typically are, a question that is answered by the variance and standard deviation of a random variable. X, find a collection X. 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 X has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined. Functions of random variableseditA new random variable g:ℝ→ℝ to the outcomes of a real-valued random variable Y is then h=g−1 exists, where y, in order to obtain y admits at most a countable number of roots (i.e., a finite, or countably infinite, number of fY(y)=∑ifX(gi−1(y))|dgi−1(y)dy| where g to be increasing. In the measure-theoretic, axiomatic approach to probability, if a random variable g:ℝ→ℝ, then Ω, since the composition of measurable functions is also measurable. (However, this is not necessarily true if (Ω,P) to Y. Example 1editLet Y=X2. y<0, then FY(y)=0ify<0. If P(X2≤y)=P(|X|≤y)=P(−y≤X≤y), so FX(x)=P(X≤x)=1(1+e−x)θ where 0"}}'>θ>0 is a fixed parameter. Consider the random variable X is a random variable with a standard normal distribution, whose density is Y=X2. We can find the density using the above formula for a change of variables: X (one positive and negative). However, because of symmetry, both halves will transform identically, i.e., dg−1(y)dy=12y. Then, fX(x)=12πσ2e−(x−μ)2/(2σ2). Consider the random variable Y has two corresponding values of x=g1,2−1(y)=±y and its derivative is X=a.s.Y) if, and only if, the probability that they are different is zero: X(ω)=Y(ω)for all ω. 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. Practical difference between notions of equivalenceeditSince we rarely explicitly construct the probability space underlying a random variable, the difference between these notions of equivalence is somewhat subtle. Essentially, two random variables considered in isolation are "practically equivalent" if they are equal in distribution -- but once we relate them to other random variables defined on the same probability space, then they only remain "practically equivalent" if they are equal almost surely. For example, consider the real random variables A, B, C, and D all defined on the same probability space. Suppose that A and B are equal almost surely (A+D=a.s.B+D, but in general 𝔼(AD)≠𝔼(CD). Therefore, two random variables that are equal in distribution (but not equal almost surely) can have different covariances with a third random variable. ConvergenceeditA 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. These are explained in the article on convergence of random variables. See alsoedit
ReferenceseditInline citationsedit
Mackey, George (July 1980). "Harmonic analysis as the exploitation of symmetry – a historical survey" (PDF). Bulletin of the American Mathematical Society. New Series. 3 (1).
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