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The population mean for a random variable and is therefore a measure of centre for the distribution of a random variable.
The expected value of random variable X is often written as E(X) or µ or µX.
The expected value is the ‘long-run mean’ in the sense that, if as more and more values of the random variable were collected (by sampling or by repeated trials of a probability activity), the sample mean becomes closer to the expected value.
For a discrete random variable the expected value is calculated by summing the product of the value of the random variable and its associated probability, taken over all of the values of the random variable.
In symbols, E(X) = Σ x P(X = x)
Example
Random variable X has the following probability function:
| x | 0 | 1 | 2 | 3 |
| P(X = x) | 0.1 | 0.2 | 0.4 | 0.3 |
E(X) = 0 x 0.1 + 1 x 0.2 + 2 x 0.4 + 3 x 0.3
= 1.9
A bar graph of the probability function, with the expected value labelled, is shown below.
See: population mean
Curriculum achievement objectives reference
Probability: Level 8
