Expectation of Function of Continuous Random Variable

Theorem

Let $X$ be a continuous random variable.

Let $\expect X$ be the expectation of $X$.

Let $g: \R \to \R$ be a real function.


Then:

$\ds \expect {g \sqbrk X} = \int_{-\infty}^\infty \map g x \map f x $


Proof


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Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): expectation (expected value)
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): expectation (expected value)
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