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Is the Expectation same as the Mean?

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If all the outcomes of an experiment are equally likely then the expectation is indeed the same as the arithmetic mean. Let us try to understand this by looking at a simple example.

Suppose that a single dice is thrown. There are six possible outcomes for the number occurring on the uppermost face of the die. All the six different outcomes – 1, 2, 3, 4, 5 and 6 – are equally likely, that is, they have an equal chance of occuring.

Let us first calculate the arithmetic mean of the six possible outcomes:

\text{Arithmetic Mean = }\frac{\text{Sum of all observations}}{\text{Number of observations}} \text{Arithmetic Mean = }\frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5

On the other hand in order to calculate the expected value we use the formula:

\text{Expected value }E(X) = \sum_{x=1}^{6} x*p(x)

Each outcome has a one in six chance of happening therefore, p(x) = 1/6 for all x.

Therefore we have that,

E(X) = \sum_{x=1}^{6} x*p(x) = 1\times \frac{1}{6} + 2\times \frac{1}{6} + 3\times \frac{1}{6} + 4\times \frac{1}{6} + 5\times \frac{1}{6} + 6\times \frac{1}{6} \text{Expected value } = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5

Therefore, we conclude that the expectation and mean are equal when the outcomes of an experiment are equally likely. The expected value is like a kind of weighted average which is equal to the simple arithmetic mean if all the weights are equal. In the above example, all the weights in the expectation were equal to 1/6.

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