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1.3.2 Expectations and population proportions

Let X be a discrete random variable which can take the values x with probabilities . Even when you do not know the value of X, you can calculate a representative value known as the expectation or expected value of X, denoted and given by

Now suppose that is a random variable that takes the values (respectively, ‘true’ and ‘false’), associated with the proposition : ‘The individual has property Q’, where i belongs to a population R of n individuals. Using (1.23), it immediately follows that the expectation of this random variable is the probability that the proposition is true:

Given that expectation is additive, the sum of probabilities ...

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