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Vs let x and y be two random variables and g x y be an

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Unformatted text preview: X , Y ) ∈ A) = ∑ pX ,Y (x , 7 − x ). x =1 M. Chen ([email protected]) ENGG2430C lecture 6 9 / 17 Expectation and Variance of Functions of R.V.s Let X and Y be two random variables, and g (X , Y ) be an arbitrary function of X and Y , E [g (X , Y )] = ∑ ∑ g (x , y )pX ,Y (x , y ) x y Var (g (X , Y )) = E g 2 (X , Y ) − E 2 [g (X , Y )]. In general E [g (X , Y )] = g (E [X ], E [Y ]), and Var (g (X , Y )) = g (Var(X ), Var(Y )) (examples) Let X and Y be two r.v.s with finite E [X ] and E [Y ] Suppose that X and Y are independent, E [X + Y ] =?E [X ] + E [Y ] Show that for any two random variables X and Y (not necessarily independent), E [X + Y ] = E [X ] + E [Y ] M. Chen ([email protected]) ENGG2430C lecture 6 10 / 17 Right or Wrong? If X and Y are independent, then E [XY ] = E [X ]E [Y ] E [X ] E [X /Y ] = E [Y ] E [g (X )h(Y )] = E [g (X )]E [h(Y )] M. Chen ([email protected]) ENGG2430C lecture 6 11 / 17 PMF and Probability Law pX (x ) ⇔ P (A) pX ,Y (x , y ) ⇔ P (A ∩ B ) pX |Y (x |y ) ⇔ P (A|B ) pX ,Y ,Z (x , y , z ) = pX (x )pY |X (y |x )pZ |X ,Y (z |x , y ) pX (x ) Bayes’law: pX |Y (x |y ) = pY |X (y |x ) pY (y ) Total probability law: pX (x ) = ∑...
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