expected value

Let us first consider a discrete random variable $X$ with values in $ℝ$. Then $X$ has values in an at most countable set $𝒳$. For $x\in 𝒳$ denote the probability that $X=x$ by $P_x$. If

$$\sum _{x\in 𝒳}|x|P_x$$

converges, the sum

$$\sum _{x\in 𝒳}x{P}_x$$

is well-defined. Its value is called the expected value, expectation or mean of $X$. It is usually denoted by $E(X)$.

Taking this idea further, we can easily generalize to a continuous random variable $X$ with probability density $\varrho$ by setting

$$E(X)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x\varrho (x)𝑑x,$$

if this integral exists.

From the above definition it is clear that the expectation is a linear function, i.e. for two random variables $X,Y$ we have

$$E(aX+bY)=aE(X)+bE(Y)$$

for $a,b\in ℝ$.

Note that the expectation does not always exist (if the corresponding sum or integral does not converge, the expectation does not exist. One example of this situation is the Cauchy random variable).

Using the measure theoretical formulation of stochastics, we can give a more formal definition. Let $(\mathrm{\Omega },𝒜,P)$ be a probability space and $X:\mathrm{\Omega }\to ℝ$ a random variable. We now define

$$E(X)={\int }_{\mathrm{\Omega }}X𝑑P,$$

where the integral is understood as the Lebesgue-integral with respect to the measure $P$.

Title	expected value
Canonical name	ExpectedValue
Date of creation	2013-03-22 11:53:42
Last modified on	2013-03-22 11:53:42
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	21
Author	mathwizard (128)
Entry type	Definition
Classification	msc 60-00
Classification	msc 46L05
Classification	msc 82-00
Classification	msc 83-00
Classification	msc 81-00
Synonym	mean
Synonym	expectation value
Synonym	expectation
Related topic	AverageValueOfFunction