# Bivariate RVs - TWO-DIMENSIONAL RANDOM VECTORS Joint...

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TWO-DIMENSIONAL RANDOM VECTORS Joint Cumulative Distribution Function (cdf) Definitions: [ ] , (,) P XY F x y X x and Y y ≤≤ Properties: 1) , (,)1 F ∞∞ = 2) ,, ( ,) (, ) 0 F yF x −∞ = 3) (,) l im (,) xa F ay F xy + = 4) [ ] 1 2 ,2 ,1 P , x X x Y y Fx y y <≤ ≤= 5) [ ] 1 21 2 2 2 1 1 P , ( , )( , , , ) x X x y Y yFx y y < < = −−+ X Y −∞ −∞

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Joint probability density function (pdf) Definitions: 2 ,, (,) XY f xy F ∂∂ Properties: 1) , (,) 0 f 2) , 1 f x y dxdy ∞∞ −∞ −∞ = ∫∫ 3) [ ] , P , db ca a X b c Y d f x y dx dy < <≤ = PHYSICAL MEANING [ ] , ( , ) and y f x y dxdy P x X x dx Y y dy = ≤+ -4 -2 0 2 4 -4 -2 0 2 4 0 0.02 0.04 X Y
Find a marginal pdf from the joint pdf , () (,) X XY f x f x y dy −∞ = or , X y p x p xy =

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Homework: continuous bi-variate random variables () , 0, 0 (,) 0 ax by XY ke x y f xy otherwise −+ >> = i) find value of k ii) find marginal pdf’s
Homework: discrete bi-variate random variables , (2 ) 1, 2; 1, 2 (,) 0 XY kxy x y p xy otherwise += = = i) find value of k ii) find marginal pmf’s

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Conditional cdf [ ] [ ] [ ] ,, | (,) P, (|) P | P () yy XY YX XX f x u x du f x u du Y yX x F yx Xx fx x −∞ −∞ ≤= = = ∫∫  Conditional pdf { } { } [ ] , || , | (|) Physcial meaning: Pr and ( ) Pr P| X X f xy d f F dy f x f xy xy Yy f yx y f xx X x = ∆∆ ∆= = = 
Moments from Bi-variate Distributions MEAN OF INDIVIDUAL RANDOM VARIABLE , [ ] () (,) X X XY E X x f x dx dx x f x dx x dy f x y ∞∞ −∞ −∞ −∞ −∞ = = = ∫∫ CONDITIONAL EXPECTATION | [| ] (|) YX E Y X x y f y x dy −∞ = = CORRELATION , E XY xy f x y dxdy −∞ −∞ = X and Y are said to be orthogonal if correlation is zero

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Bivariate RVs - TWO-DIMENSIONAL RANDOM VECTORS Joint...

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