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Unformatted text preview: 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 ( x ) is some function. Then, E ( ( X )) = integraldisplay - ( x ) f ( x ) dx. There is also a discrete analogue of this identity. Suppose that X 1 , ..., X k are random variables and that 1 , ..., k are real numbers. Then, E ( 1 X 1 + + k X k ) = 1 E ( X 1 ) + + 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 ) ....
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