Expected Value

The weighted average of some function $f(x)$ under a probability distribution $p(x)$ is called the expectation or expected value of $f(x)$ and defined by $E(f)$.

For a discrete distribution, this is given by summing over all possible values of $x$ in the form

$$E[f]=\sum _{x}p(x)f(x)$$

where the average is weighted by the relative probabilities of the different values of $x$.

For continuous distributions, expectations are expressed in terms of integration with respect to the corresponding Probability Density Function:

$$E[f]=\int p(x)f(x)dx$$

Approximation from Sample

If we are given a finite number $N$ of points drawn from the probability distribution/density, then the expectation can be approximated as a finite sum over these points:

$$E[f]\approx \frac{1}{N}\sum _{n=1}^{N}f(x_n)$$

This approximation becomes exact in the limit as $N\to \infty$.

Multivariate Expectation

For functions of several variables, we can use a subscript to indicate which variable is being averaged over, so that for instance

$$E_x[f(x,y)]$$

denotes the average of the function $f(x,y)$ with respect to the distribution of $x$. Note that $E_x[f(x,y)]$ will be a function of $y$.

Conditional Distribution

We can also consider a conditional expectation with respect to a conditional distribution, so that

$$E_x[f|y]=\sum _{x}p(x|y)f(x)$$

which is a function of $y$.

For continuous variables, this conditional expression becomes

$$E_x[f|y]=\int p(x|y)f(x)dx$$