Random variable

In probability theory, a branch of mathematics, a random variable is, as its name suggests, a variable that can take on random values. More formally, it is not actually a variable , but a function whose argument takes on a particular value according to some probability measure (a measure that takes on the value 1 over the largest set on which it is defined).

Formal definition

Let $(\Omega ,{\mathcal {F}},P)$ be an arbitrary probability space and $(\Omega ',{\mathcal {F}}')$ an arbitrary measurable space. Then a random variable is any measurable function X mapping $(\Omega ,{\mathcal {F}})$ to $(\Omega ',{\mathcal {F}}')$.

An easy example

Consider the probability space $(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ),P)$ where ${\mathcal {B}}(\mathbb {R} )$ is the sigma algebra of Borel subsets of $\mathbb {R}$ and P is a probability measure on $\mathbb {R}$ (hence P is measure with $P(\mathbb {R} )=1$). Then the identity map $I:(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))\rightarrow (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))$ defined by $I(x)=x$ is trivially a measurable function, hence is a random variable.

References

1. P. Billingsley, Probability and Measure (2 ed.), ser. Wiley Series in Probability and Mathematical Statistics, Wiley, 1986.
2. D. Williams, Probability with Martingales, Cambridge : Cambridge University Press, 1991.

External links

1. Probability tutorial at Probability.net