= Expected Values =

An '''expected value''' is the estimated outcome of an event. The math notation is ''E[X]''.

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== Evaluation ==

For a discrete distribution, the expected value of ''X'' is generally ''Σ x P(x)'' (for all ''X=x'').

For a continuous distribution, the expected value of ''X'' is generally ''∫ x P(x) dx'' (for all ''X=x'').



=== Bernoulli ===

For a [[Statistics/BernoulliDistribution|Bernoulli-distributed]] variable (taking value 1 with probability ''p'' and value 0 with probability ''1-p''), the expected value is ''p''.

''E[X] = Σ x P(x)''

''E[X] = (0) P(0) + (1) P(1)''

''E[X] = P(1)''

''E[X] = p''

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== Linearity ==

Expectations are linear.

''E[X + c] = E[X] + c''

''E[X + Y] = E[X] + E[Y]''

''E[aX] = a E[X]''

''E[aX + bY] = a E[X] + b E[Y]''



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