Vs x and y are independent if for any values x and y

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Unformatted text preview: pY (y )pX |Y (x |y ) y = ∑ pX ,Y (x , y ) y Independence: M. Chen ([email protected]) ? ENGG2430C lecture 6 12 / 17 Independence of R.V.s Definition: Two r.v.s X and Y are independent if for any values x and y pX ,Y (x , y ) = pX (x )pY (y ), or equivalently pX |Y (x |y ) (P (A ∩ B ) = P (A)P (B )) P (X = x |Y = y ) = P (X = x ) pX ( x ) . Knowing the values of X reveals no information on the values of Y The two experiments whose outcomes are modeled by X and Y are independent Right or wrong: Two Bernoulli r.v.s X and Y are independent if pX ,Y (1, 1) M. Chen ([email protected]) = pX (1)pY (1). ENGG2430C lecture 6 13 / 17 Independence Simplifies Computation Let X1 , X2 , . . . Xn be n independent random variables. Let Y = ∑n=1 Xi , then we have i n E [Y ] = ∑ E [Xi ] (= n · µ if Xi has the same expectation µ ) i =1 n Var (Y ) = ∑ Var (Xi ) = n · σ 2 if Xi has the same variance σ 2 i =1 Let X and Y be two independent random variables, prove that E [XY ] = E [X ]E [Y ] Covariance of two random variables X and Y is defined as Cov (X , Y ) = E [(X − E [X ]) (Y − E [Y ])] = E [XY ] − E [X ]E [Y ] If X and Y are independent, then Cov (X , Y ) = 0 M....
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