#### Expectation

The expectation or expected value of a random variable is denoted by . It can be considered as a kind of weighted average for , in which the weights are obtained from the probability distribution. If is discrete, then

 (9.10)

If is continuous, then9.2

 (9.11)

One can then define conditional expectation, which applies a given condition to the probability distribution. For example, if is discrete and an event is given, then

 (9.12)

Example 9..5 (Tossing Dice)   Returning to Example 9.4, the elements of are already real numbers. Hence, a random variable can be defined by simply letting . Using (9.11), the expected value, , is . Note that the expected value is not necessarily a value that is expected'' in practice. It is impossible to actually obtain , even though it is not contained in . Suppose that the expected value of is desired only over trials that result in numbers greater then . This can be described by the event . Using conditional expectation, (9.12), the expected value is .

Now consider tossing two dice in succession. Each element is expressed as in which . Since , the random variable needs to be slightly more interesting. One common approach is to count the sum of the dice, which yields for any . In this case, .

Steven M LaValle 2012-04-20