EE 5345 Multiple Random Variables 2-D RV’s Conditional RV’s

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EE 5345 Multiple Random Variables 2-D RV’s Conditional RV’s © Robert J. Marks II

Two-Dimensional Random Variables X and Y are joint random variables. They are described by the joint cumulative distribution function FXY(x,y) = Pr[X x , Y  y] “AND” © Robert J. Marks II

Properties of 2D CDF CDF: FXY(x,y) = Pr[X x , Y  y] 0  FXY(x,y)  1 FXY(,)= Pr[X  , Y  ] = 1 FXY(-,-)= Pr[X - , Y  -] = 0 © Robert J. Marks II

Properties of 2D CDF FXY(x,)= Pr[X x , Y  ] = Pr[X x ] = FX(x)  Marginal CDF FXY(x,y)  Joint CDF © Robert J. Marks II

Properties of 2D CDF (cont) If X and Y are independent FXY(x,y) = Pr[X x , Y  y] = Pr[X x] Pr[ Y  y] = FX(x) FY(y) FY(y)  Marginal CDF FXY(x,y)  Joint CDF © Robert J. Marks II

Properties of 2D CDF (cont) When x > 0 and y > 0, FXY(x+x,y+ y)  FXY(x,y) x x+x y+y y © Robert J. Marks II

Properties of 2D CDF (cont) FXY(x+x, y+ y) =Pr[X  x+x, Y  y+ y]  FXY(x, y) =Pr[X x, Y  y] x x+x y+y y © Robert J. Marks II

Properties of 2D pdf’s Total volume of 2D pdf is one. © Robert J. Marks II

Properties of 2D pdf’s (cont) If X and Y are independent Joint pdf   Marginal pdf’s © Robert J. Marks II

Histograms & 2D pdf’s: 3681 trucks weighed & measured TONS 2-D histogram Length Marginal L E N G T H Weight Marginal As in 1D, normalized histograms are empirical pdf’s. See why they are called marginals? © Robert J. Marks II

Velocity in a (2D) Gas X and Y = x and y velocities in an ideal gas. 1. X and Y should be independent: 2. Velocity should be the same in all directions: 3. Only one solution: © Robert J. Marks II

Properties of 2D pdf’s (cont) A y y x A A x © Robert J. Marks II

Properties of 2D pdf’s (cont) y Marginal – integrate out the unwanted variable… A  © Robert J. Marks II x

Example A x y A x=y © Robert J. Marks II

Example A y A z x z © Robert J. Marks II

2-D Dirac Deltas & Point Masses y0 y x x0 (x-x0)  (y-y0)=? y x0  (x-x0) y0  (y-y0) x © Robert J. Marks II

2-D Dirac Deltas (cont) Q: What is the volume of (x-x0)  (y-y0)? A: © Robert J. Marks II

2-D Discrete pdf’s jk pXY(xj,yk) =1 Probability Mass Function: pXY(xj,yk)=Pr[X=xj,Y=yk] jk pXY(xj,yk) =1 If events are independent: pXY (xj,yk)= pX (xj) pY (yk) Discrete RV’s of lattice type: pXY(i,k)=Pr[X=i,Y=k] © Robert J. Marks II

fXY(x,y)= jk pXY(xj,yk) (x-xj)  (y-yk) 2-D Discrete pdf’s fXY(x,y)= jk pXY(xj,yk) (x-xj)  (y-yk) y x © Robert J. Marks II

2-D Discrete Lattice pdf’s fXY(x,y)= jk pXY(xj,yk) (x-j)  (y-k) © Robert J. Marks II

Line Masses X = Length of a side of a cube ~ fX(x) Y = Cube volume y=x3 y=x3 y x fXY(x,y) … a pdf? © Robert J. Marks II

Conditional cdf’s & pdf’s Conditional PDF : © Robert J. Marks II

Recall the 3681 trucks: pdf <= normalized histogram pdf given A: 1.5  tons  2.0 9.0  length 10.0 TONS L E N G T H conditional pdf has same shape within the given region, A ; is zero otherwise ; and is normalized (using a different number) to give a total volume of one. © Robert J. Marks II

y y A A x x Conditional pdf has same shape within the given region, A ; is zero otherwise ; and is normalized (using a different number) to give a total volume of one. x y A y x A © Robert J. Marks II

has same shape within the given region, A ; is zero otherwise ; Conditional pdf has same shape within the given region, A ; is zero otherwise ; and is normalized (using a different number) to give a total volume of one. © Robert J. Marks II