Random variable

A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. For example, a random variable can be used to describe the process of rolling a fair die and the possible outcomes { 1, 2, 3, 4, 5, 6 }. Another random variable might describe the possible outcomes of picking a random person and measuring their height.

Definitions

Random variables

Some consider the expression "random variable" a misnomer: a random variable is a function mapping from events to numbers. Let A be a σ-algebra and Ω be 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 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

Related Topics:
σ-algebra - Event - Power set

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::X(omega) = egin{cases}0,& omega = exttt{H},\1,& omega = exttt{T}.end{cases}

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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.

Related Topics:
Measurable function - Probability space - Measurable space - Borel σ-algebra

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Let (Ω, A, P) be a probability space. Formally, a function X: Ω → R is a (real-valued) random variable if for every subset Ar = { ω : X(ω) ≤ r } where r ∈ R, we also have Ar ∈ A.

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The importance of this technical definition is that it allows us to construct the distribution function of the random variable.

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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 inOmega : X(s) > 2 } which is often written as P(X > 2) for short.

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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

Related Topics:
Probability distribution - Cumulative distribution function

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:F_X(x) = operatorname{P}(X le x)

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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 dF on R.

Related Topics:
Probability density function - Measure-theoretic

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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.

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