Expectation (math)

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The mathematical expectation of a continuously distributed random variable X with probability density function f(x) is


\mbox{E}[X]
=\int\limits_{-\infty}^\infty
x
f(x)dx.

The expectation is also known as the mean of X.

The expectation with respect to some function g(X) where X is distributed according to f(x) is


\mbox{E}[g(X)]
=\int\limits_{-\infty}^\infty
g(x)
f(x)dx.

For a discretely distributed random variable X with probability mass function pk it is

E[X] = pkxk.
k
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