The expectation of a function with respect to a probability distribution is denoted as . It represents the weighted average value of over the distribution . Often, the subscript is omitted when the distribution is clear from context.

For a discrete random variable , the expectation is defined as the probability-weighted sum of all possible values in the random variable’s support,

For continuous random variables, the expectation is defined as the integral of the product of the function and the probability density function over the entire support of the random variable:

Properties

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