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Bayesian Networks for Probabilistic Inference and Decision Analysis in Forensic Science, 2nd Edition by Colin Aitken, Paolo Garbolino, Silvia Bozza, Alex Biedermann, Franco Taroni

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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 c01-math-0478. 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 c01-math-0481 and given by

Now suppose that c01-math-0483 is a random variable that takes the values c01-math-0484 (respectively, ‘true’ and ‘false’), associated with the proposition c01-math-0485: ‘The individual c01-math-0486 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:

equation

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

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