e xy e x e y The proof in the discrete case is analogous This shows why independence of X and Y implies that E XY E X E Y The converse does not necessarily hold that is we can come up with examples of random variables X Y with E XY E X E Y but which are not independent
In general we have mathbb E XY mathbb E mathbb E Y mid X X which tells that expectation of XY can be computed by first averaging Y over given information of X and then taking unconditional average X E X and Y E Y and k be a positive integer 1 The kth moment of X is de ned as E Xk If k 1 it equals the expectation 2 The kth central moment of X is de ned as E X X k If k 2 then it is called the variance of X and is denoted by var X The positive square root of the variance is called the standard deviation 3
e xy e x e y
e xy e x e y
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E XY E X E Y Laws Of Expectation YouTube
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Probability mathbb E XY mathbb E X mathbb E Y Where Do
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Covariance formula E XY E X E Y or expectation of product minus product of expectations is frequently useful Note if X and Y are independent then Cov X Y 0 I 1x i y i approaches the expectation E XY For example if X is height and Y is weight E XY is the average of height weight We are interested in E XY because it is used for calculating the covariance and correlation which are measures of how closely related X and Y are see Section 3 2 Properties of Expectation
Why is E XY E XE Y X Is this using the properties of conditional expectation and is there a general formula that can be applied when you have E E E Y X From the proof X Y 1 i X E X X Y E Y Y equality with probability 1 i e i X E X is a linear function of Y E Y In general X Y is a measure of how closely X E X can be approximated
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Let X and Y be discrete random variables Prove the linearity of expectation E X Y E X E Y An exercise problem in probability theory The solution is given E X E Y 0 To measure the spread of a random variable X that is how likely it is to have value of X very far away from the mean we introduce the variance of X denoted by var X Let us consider the distance to the expected value i e jX E X j
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e xy e x e y - Covariance formula E XY E X E Y or expectation of product minus product of expectations is frequently useful Note if X and Y are independent then Cov X Y 0