Expected value

In statistics, the sample mean is the sum of all the data values divided by the number of data values in the population. Its formula is given by:

$$\bar{x}=\frac{1}{n}\sum _{k=0}^{\infty }{f}_i{x}_i$$

If we’re dealing with the whole population of data, we can also represent the mean with $\mu$.

Properties

The expected value also has several helpful properties. For a constant $c$:

$$E(c)={\int }_{-\infty }^{\infty }c{f}_X(x)dx=c$$ $$E(cX)={\int }_{-\infty }^{\infty }c{f}_X(x)dx=cE(X)$$

And we can break up the expected value of sums, into a sum of expected values:

$$E(Y)=E(\sum _{k=1}^n{g}_k(X))=\sum _{k=1}^{n}E(g_k(X))$$ $$E(X+Z)=E(X)+E(Z)$$

Probability distributions

For probability distributions, we refer to the mean as the expected value. We keep the same summation principle. We can think of $E(X)$ as the centre-of-mass, if $f_X(x)$ is a distribution of mass on the real line.

In the continuous case:

$$E(X)={\int }_{-\infty }^{\infty }t{f}_X(t)dt$$

In the discrete case:

$$E(X)=\sum _k{x}_k{p}_X(x_k)$$

The expected value is only defined if the integral or sum converges absolutely. There are certain RVs for which they don’t converge (Zipf, Pareto, Cauchy).

We also define the $n$th moment of the random variable $X$ as:

$$E(X^n)={\int }_{-\infty }^{\infty }{x}^n{f}_X(x)dx$$

We define the conditional expected value as:

$$E(X|A)={\int }_{-\infty }^{\infty }x{f}_X(x|A)dx$$

Multiple random variables

If the random variables are pairwise independent, then:

$$E(XY)=E(X)E(Y)$$

The conditional expected value of a random variable $Y$ given $X=x$ is:

$$E(Y|X=x)={\int }_{-\infty }^{\infty }y{f}_Y(y|x)dy$$

The expected value of a multi-RV function $Z=g(X,Y)$ is given by:

$$E(Z)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }g(x,y)f_{X,Y}(x,y)dxdy$$ $$E(Z)=\sum _i\sum _{n}g(x_i,y_n)p_{X,Y}(x_i,y_n)$$

In the multiple RV case, the sums break apart as in the single RV case.

The $jk$th joint moment of $X$ and $Y$ is given by:

$$E(X^j{Y}^k)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }$$

See also

• Variance
• Standard deviation