Definition:Expectation/Continuous

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Definition

Let $X$ be a continuous random variable over the probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $F = \Pr \paren {X < x}$ be the cumulative probability function of $X$.


The expectation of $X$ is written $\expect X$, and is defined over the probability measure as:

$\expect X := \displaystyle \int_{x \mathop \in \Omega} x \rd F$

whenever the integral is absolutely convergent, i.e. when:

$\displaystyle \int_{x \mathop \in \Omega} \size x \rd F < \infty$


Also, from the definition of probability density function $f_X$ of $X$, we see it can also be written over the sample space:

$\displaystyle \expect X := \int_{x \mathop \in \Omega_X} x \ f_X \paren x \rd x$


Also known as

The expectation of $X$ is also called the expected value of $X$ or the mean of $X$, and (for a given continuous random variable) is often denoted $\mu$.


The terminology is appropriate, as it can be seen that an expectation is an example of a normalized weighted mean.

This follows from the fact that a probability density function is a normalized weight function.


Also see

It can also be seen that the expectation of a continuous random variable is its first moment.


Technical Note

The $\LaTeX$ code for \(\expect {X}\) is \expect {X} .

When the argument is a single character, it is usual to omit the braces:

\expect X


Sources