Slide_03+--+Multiple+Random+Variables

Slide_03+--+Multiple+Random+Variables - A 2-D Random...

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A 2-D Random Variable or Two Random Variables: {} , ,, , , The joint pdf: ( , ), of non-negative values, defined over a 2-D region of definition (ROD), The joint cdf: ( , ) ( , ) , with ( , ) 1 ,( , ) XY y x fx y F x y f u v du dv f PX xY y F xy +∞ +∞ −∞ −∞ −∞ −∞ == ≤≤ = ∫∫ Figure 4.1 (p. 154) The outcome of the 2-D integration over the shaded area of the ( X,Y ) plane corresponds to the joint cumulative distribution function (cdf) , (, ) F xy
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, (, ) XY x yS , () P r o b { (,) } (,) B P B x y B f u v du dv =∈ = ∫∫ , x Figure 4.2 (p. 157) Subsets B of the ( X,Y ) plane. Points ( X,Y ) S X,Y are marked by bullets. Other Cases: , () { ( ,) } ( , ) B PB P xy B f uvdudv =
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Finding the joint CDF from the joint pdf – integrating pdf over the regions defined by the nominal point (x,y) Five cases for the CDF F XY ( x,y ) calculation, Fig4.3 (p. 163) , 2 ,, 20 1 Given the joint pdf ( , ) 0o t h e r w i s e 1 Case-a: ( , ) 0 Case-b: ( , ) 2 ( ) 2 XY yx fx y F x y Fx yy y x y ≤≤≤ =  == +  
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Figure 4.3.2 (cont’d) p. 163 The graph for finding the CDF Fx ( x ), case-c and case-d 22 2 , , 11 Case-c: ( , ) 2 ( ) Case-d: ( , ) 2 (1 ) XY Fx y x x y yy y  =⋅ = +  
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Figure 4.3.2
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Slide_03+--+Multiple+Random+Variables - A 2-D Random...

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