Expected Value

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The expected value of a variable is the long-run average value of that variable. The expected value of a statistic is therefore the mean of the sampling distribution of the statistic.

If the expected value of a statistic is the parameter the statistic is estimating, the statistic is an unbiased estimate of the parameter.

Expected values of variables are indicated by an "E" with the variable enclosed in brackets. Thus, E[X] is read as the expected value of X.

Some basic rules of expected values are shown below:

1. E[X] = μ where μ is the mean of X.
2. σ² = E[X - μ]² where σ² is the variance of X and μ is the mean of X.
3. E[X]² = σ² + μ²
4. E[X + Y] = E[X] + E[Y]
5. E[XY] = E[X]E[Y] if X and Y are independent.
6. In general, E[X/Y] does not equal E[X]/E[Y]