Random Variable

Unlike the common practice with other mathematical Variable s, a random variable cannot be assigned a value; a random variable does not describe the actual outcome of a particular experiment, but rather describes the possible, as-yet-undetermined outcomes in terms of real numbers. Although such simple examples as rolling a die and measuring heights allow easy visualisation of the practical use of random variables, their mathematical construction allows mathematicians the convenience of dealing with much Measure-theoretic Probability Theory in the more familiar domain of Real -valued functions. Conversely, the concept also places experiments involving real-valued outcomes firmly within the measure-theoretic framework. DEFINITIONS Random variables Some consider the expression ''random variable'' a Misnomer , as a random variable is not a Variable but rather a Function that maps Events to numbers. Let ''A'' be a σ-algebra and Ω the space of events relevant to the experiment being performed. In the die-rolling example, the space of events is just the possible outcomes of a roll, i.e. Ω = { 1, 2, 3, 4, 5, 6 }, and ''A'' would be the Power Set of Ω. In this case, an appropriate random variable might be the Identity Function ''X''(ω) = ω, such that if the outcome is a '1', then the random variable is also equal to 1. An equally simple but less trivial example is one in which we might toss a coin: a suitable space of possible events is Ω = { H, T } (for heads and tails), and ''A'' equal again to the power set of Ω. One among the many possible random variables defined on this space is :: $X(\omega) = \begin{cases}0,& \omega = exttt{H},\1,& \omega = exttt{T}.\end{cases}$ Mathematically, a random variable is defined as a Measurable Function from a Probability Space to some Measurable Space . This measurable space is the space of possible values of the variable, and it is usually taken to be the real numbers with the Borel σ-algebra . This is assumed in the following, except where specified. Let (Ω, ''A'', ''P'') be a probability space. Formally, a function ''X'': Ω → R is a (real-valued) ''random variable'' if for every subset A_''r'' = { ω : ''X''(ω) ≤ ''r'' } where ''r'' ∈ R, we also have A_''r'' ∈ ''A''. The importance of this technical definition is that it allows us to construct the distribution function of the random variable. Distribution functions If a random variable $X: \Omega o \mathbb{R}$ defined on the probability space $(\Omega , P)$ is given, we can ask questions like "How likely is it that the value of $X$ is bigger than 2?". This is the same as the probability of the event $\{ s \in\Omega : X(s) > 2 \}$ which is often written as $P(X > 2)$ 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 Function : $F_X(x) = \operatorname{P}(X \le x)$ and 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, and sometimes to construct them. 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. FUNCTIONS OF RANDOM VARIABLES If we have a random variable ''X'' on Ω and a Measurable Function ''f'': R → R, then ''Y'' = ''f''(''X'') will also be a random variable on Ω, since the composition of measurable functions is also measurable. The same procedure that allowed one to go from a probability space (Ω, P) to (R, dF_''X'') can be used to obtain the distribution of ''Y''. The cumulative distribution function of ''Y'' is : $F_Y(y) = \operatorname{P}(f(X) \le y).$ Example Let ''X'' be a real-valued, Continuous Random Variable and let ''Y'' = ''X''². Then, : $F_Y(y) = \operatorname{P}(X^2 \le y).$ If ''y'' < 0, then P(''X''² ≤ ''y'') = 0, so : $F_Y(y) = 0\qquad\hbox{if}\quad y < 0.$ If ''y'' ≥ 0, then
	:<math>d(X,Y)	\sup_x\operatorname{P}(X \le x) - \operatorname{P}(Y \le x),</math>
	:<math>\operatorname{E}(X-Y^p)	0</math>
	:<math>d P(X, Y)	\operatorname{E}(X-Y^p)</math>
	:<math>d \infty(X,Y)	\sup_\omegaX(\omega)-Y(\omega),</math>

	CATEGORIES ABOUT RANDOM VARIABLE

	probability and statistics

	probability theory

	randomness

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Information About

Random Variable