# Expected Value

The expected value is the "weighted mean" of all the possible values.

This is the sum of each value multipled by the probability of it occurring.

This is one type of location parameter.

The expected value is the value you would expect to get on average

The expected value of a distribution is symbolised by E(X) or µ.

Stating the expected value gives a general impression of the behaviour of some random variable without giving full details of its probability distribution (if it is discrete) or its probability density function (if it is continuous).

Two random variables with the same expected value can have very different distributions.

Example

Discrete case : When a die is thrown, each of the possible faces 1, 2, 3, 4, 5, 6 (the xi's) has a probability of 1/6 (the p(xi)'s) of showing. The expected value of the face showing is therefore:

µ = E(X) = (1 x 1/6) + (2 x 1/6) + (3 x 1/6) + (4 x 1/6) + (5 x 1/6) + (6 x 1/6) = 3.5

Notice that, in this case, E(X) is 3.5, which is not a possible value of X.

If X is a discrete random variable with possible values x1, x2, x3, ..., xn, and p(xi) denotes P(X = xi), then the expected value of X is defined by:

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