When we are dealing with a value which is not precisely known, it is often possible and useful to ask the question, "What do we expect the value to be?" The answer to this question is "the Expectation Value."

The expectation value is a kind of weighted average of all the possible values, where more probable values are given a greater weight and less probable values are given a smaller weight.

The expectation value for some variable is often symbolized by placing the variable between angled brackets, <>. Thus the expectation value for some variable or function, X can be denoted as <X>.

The formula for the expectation value is written as follows:

<X> =	∑	X(s)p(s)
	s∈S

Here, S is a set of all possible states of the system being considered, p(s) is the probability that the system will be in state s, and X(s) is the value of X when the system is in state s.

If we were to consider a standard six sided die being rolled, then there would be 6 possible final states for the final system. If we wanted the expectation value for the final number on the die, we would calculate it as follows.

<X> =	6
	∑	(1/6)s = (1/6)⋅(1+2+3+4+5+6) = 21/6 = 3.5
	s=1

The Expectation Value can also be expressed as an integral for continuous systems, rather than discrete systems.

<X> = ∫x⋅f(x)dx

Here f(x) is a probability distribution and the integral ∫f(x)dx is unitary, meaning, that it must equal 1.

The expectation value is of great importance not only in the mathematical field of statistics, but also in the field of Quantum Mechanics, where the Heisenberg Uncertainty Principle holds that physical properties such as position and momentum cannot be known exactly at the same time. A particles position or momentum is very seldom known very well and so the expectation value is very frequently relied upon in order to make determinations about where to look for a particle, or to give physical meaning to the wave equations which are used to represent them.