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