# V with a nite variance varx and let and be two

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Unformatted text preview: quadratic operation on x ’s. Var(X ) E (X − E [X ])2 = ∑ (x − E [X ])2 pX (x ) x Let X be an r.v. with a ﬁnite variance Var[X ], and let α and β be two constants: Var[α ] =? 0 Var[α X ] =? α 2 Var[X ] Var[α X + β ] =? α 2 Var[X ] Let Y = g (X ) where g (·) is a function, compute Var(Y ). Var(Y ) =E Y 2 − E 2 [Y ] 2 = ∑ g 2 (x )pX (x ) − x M. Chen ([email protected]) ENGG2430C lecture 6 ∑ g (x )pX (x ) x 7 / 17 Joint PMF for Multiple Random Variables Experiments often involve several random variables Values of several stocks: how are they coupled? Packets traverse n Internet routers: what is the total transmission delay? The joint PMF of two r.v.s X and Y is deﬁned as pX ,Y (x , y ) P (X = x , Y = y ) . P ((X , Y ) ∈ A) = ∑ pX ,Y (x , y ). (x ,y )∈A Marginal PMF: P (X = x ) = ∑ P (X = x , Y = y ) = ∑ pX ,Y (x , y ) y y The conditional PMF of X given that Y = y is expressed as: pX |Y (x |y ) = P (X = x |Y = y ) = M. Chen ([email protected]) ENGG2430C lecture 6 pX ,Y (x , y ) . pY (y ) 8 / 17 Example: Uniform Joint PMF Throw two dies once. Let a r.v. X be the outcome of the ﬁrst die, and a r.v. Y be the outcome of the second die. pX ,Y (x , y ) = 1 36 . Event A = {X + Y = 7}: 6 P ((...
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