Expected Value
The expected value of a random variable is its average, i.e. the value that can be assumed for a finite set of observations without impacting the sum.
Description
For a random variable X, the expected value is usually notated as E[X], E(X), X̅.
More commonly though, a particular distribution function is assumed and the expected value is written in terms appropriate for the corresponding first moment. For example:
for a Normal distribution, μ
for a Bernoulli distribution, p
for a Binomial distribution, np
for a Poisson distribution, λ
for a Uniform distribution, (a+b)/2
- and so on...
Formulation
Formally, the first moment of a continuous quantitative (numeric) random variable is:
where Ω is the sample space and f is the probability density function for that distribution.
And for a discrete quantitative random variable:
where Ω is the set of possible values and f is the probability mass function for that distribution (although the distributions of discrete random variables are commonly given as tables, rather than mathematical functions).
But when working with a set of observations X, where for each observation i there is an observed value xi, and for which there is a function p describing the probability of X taking on the value xi (sometimes expressed as p(X=x)), the most common formula is:
Expected Value as a Function
Expectations are linear.
If the expected value of a random variable X is known, adding a constant to X also adds that constant to the expected value. Multiplying X by a constant factor also multiplies the known variance by that factor. Succinctly: E[aX + b] = a E[X] + b.
The expected value of the sum of two random variables X and Y is the sum of their expected values: E[X + Y] = E[X] + E[Y].
Similarly, if two random variables are independent, then the expected value of their product is the product of their expected values: E[XY] = E[X]E[Y]. This is not true if the two are related; refer to the common formula for covariance: Cov(X, Y) = E[XY] - E[X]E[Y].
Also consider the ratio of two independent random variables: E[X/Y] = E[X]E[1/Y]. Note that this is not equivalent to E[X]/E[Y].