Definition:Expectation/Continuous
Definition
Let $X$ be a continuous random variable over the probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $F = \map \Pr {X < x}$ 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:
- $\displaystyle \expect X := \int_{x \mathop \in \Omega_X} x \, \map {f_X} x \rd x$
Also known as
The expectation of a random variable $X$ is also called the expected value of $X$ or the mean of $X$, and (for a given 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 mass function is a normalized weight function.
Various forms of $E$ can be seen to denote expectation:
- $\map E X$
- $\map {\mathrm E} X$
- $E \sqbrk X$
- $\mathop {\mathbb E} \sqbrk X$
and so on.
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