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Countable Riemann Multiple Integral Set Finite Operations On Relations Real Number Measure Random Event Distribution Function Borel Space Discrete Measurable Functions Absolutely Continuous Function

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Random Variable

If $(\Omega,\mathcal {A},P)$ is a probability space, then a random variable on $\Omega$ is a measurable function $X: (\Omega,\mathcal {A}) \to S$ to a measurable space (frequently taken to be the real numbers with the standard measure). The law of a random variable is the probability measure $PX^{-1}:S\to \mathbb{R}$ defined by $PX^{-1}(s)=P(X^{-1}(s))$ .

A random variable is said to be discrete if the set $\{X(\omega) : \omega \in \Omega \}$ (i.e. the range of ) is finite or countable. A more general version of this definition is as follows: A random variable is discrete if there is a countable subset of the range of such that $P(X \in B)=1$ (Note that, as a countable subset of $\mathbb{R}$ , is measurable).

A random variable is said to be continuous if it has a cumulative distribution function which is absolutely continuous.

Example:

Consider the event of throwing a coin. Thus, $\Omega = \{ H, T \}$ where is the event in which the coin falls head and the event in which falls tails. Let number of tails in the experiment. Then is a (discrete) random variable.

1	Riemann Multiple Integral
1	Measure
1	Real Number
1	Operations On Relations
1	Finite
1	Countable
1	Set
2	Random Event
2	Distribution Function
2	Borel Space
2	Measurable Functions
2	Discrete
6	Absolutely Continuous Function