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expectation_notes

# expectation_notes - Some facts about expectation 1...

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Some facts about expectation October 22, 2010 1 Expectation identities There are certain useful identities concerning the expectation operator that I neglected to mention early on in the course. Now is as good a time as any to talk about them. Here they are: Suppose that X is a continuous random variable having pdf f ( x ), and suppose that α ( x ) is some function. Then, E ( α ( X )) = integraldisplay -∞ α ( x ) f ( x ) dx. There is also a discrete analogue of this identity. Suppose that X 1 , ..., X k are random variables and that λ 1 , ..., λ k are real numbers. Then, E ( λ 1 X 1 + · · · + λ k X k ) = λ 1 E ( X 1 ) + · · · + λ k E ( X k ) . This property is called “Linearity of Expectation”. Suppose that X 1 , ..., X k are independent random variables. Then, the expectation of a product is the product of expectations; that is E ( X 1 · · · X k ) = E ( X 1 ) E ( X 2 ) · · · E ( X k ) . Note: You really do need independence here. There are examples of r.v.’s that are not independent, for which this identity fails to hold.

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expectation_notes - Some facts about expectation 1...

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