Probability-speak for average value (of a random variable). This is the weighted average of its values, where the weight of each value is the probability that the random variable take on that value.

The extension to continuous and other variables is the usual boring extension to integrals (and density functions).

The expectation value is a value applied to a probability distribution that roughly corresponds to what we would think of as the average. Given that you have some process with a probability distribution, if you were to do that process many times and average the outcomes that mean value would tend toward the expectation value of the probability distribution as the number of trials increased toward infinity. Formally, given a distribution of probabilities for a set of outcomes x(i) with probability P(x(i)), the expectation value of a function f(x) for that probability distribution, denoted <f>, would be the sum of the values f(x(i)) weighted by the probabilities P(x(i)).

The expectation value may also be denoted in others ways including f with a bar above or exp(f) (which I know is like the way people express e, sometimes, but people use it) and others. There are a few important things to note about the expectation value. For a finite distribution there will be a finite set of possible values of a given function f, but the expectation value does not have to be the value of f for one of the outcomes actually possible for an individual trial, since it represents the average of multiple outcomes. Also, if all outcomes are equally likely (for n possible outcomes, P=1/n) then the expectation value simply becomes the arithmetic mean of f over all outcomes. Finally, I think an example is in order.

#### Example

Let's take the example of rolling a pair of dice. If you roll a pair of 6 sided dice, the possible outcomes for the total amount showing are: 2,3,4,5,6,7,8,9,10,11,12. Not too interesting so far. The number of ways, N, you can get each outcome, s, is as follows:

```s       N
-----------
2   |   1
3   |   2
4   |   3
5   |   4
6   |   5
7   |   6
8   |   5
9   |   4
10  |   3
11  |   2
12  |   1
```

The total number of possible combinations of the values on each die (not the sum) is 36, so the probability of each outcome for the sum, s, is N/36. Now we can look at some expectation values. Let's calculate <s>. We'll leave the factor of 36 to the end and divide through by it.
<s> = (2*1+3*2+4*3+5*4+6*5+7*6+8*5+9*4+10*3+11*2+12*1)/36=252/36=7

So, if you roll two properly weighted dice lots of times and average the values you get for the total amount showing, the resulting mean would be close to 7. The more you roll the dice, the less likely that your value will differ much from 7. Like I said, all this counts on you having properly weighted dice, which many are not (have you ever noticed that about the dice that come with Risk?). If, however, you calculate the expectation value of the function s^2, then:
<s^2>=Sum of (s^2)*P(s)=1974/36=54.8333... Notice, that that is not the value of s^2 for any of the possible values of s.

The concept of the expectation value is important in many fields that use statistics, including statistical mechanics and quantum mechanics.

Note: The above example, while simple, was just pulled out of my ass as I write this, so let me know if I messed up the math. Also, please /msg me and let me know if I made any other mistakes or if you find something unclear.

Ex`pec*ta"tion (?) n. [L. expectio. exspectio: cf. F. expectation.]

1.

The act or state of expecting or looking forward to an event as about to happen.

"In expectation of a guest."

Tennyson.

My soul, wait thou only upon God, for my expectation is from him. Ps. lxii. 5.

2.

That which is expected or looked for.

Why our great expectation should be called The seed of woman. Milton.

3.

The prospect of the future; grounds upon which something excellent is expected to happen; prospect of anything good to come, esp. of c or rank.

His magnificent expiations made him, in the opinion of the world, the best much in Europe. Prescott.

By all men's eyes a youth of expectations. Otway.

4.

The value of any chance (as the prospect of prize or property) which depends upon some contingent event. Expectations are computed for or against the occurrence of the event.

5. Med.

The leaving of the disease principally to the efforts of nature to effect a cure.

Expectation of life, the mean or average duration of the life individuals after any specified age.

Syn. -- Anticipation; confidence; trust.