Transcription of Joint and Marginal Distributions
1 Joint and Marginal DistributionsOctober23,2008We will now consider more than one random variable at a time. As we shall see, developing the theoryofmultivariatedistributions will allow us to consider situations that model the actual collection of dataand form the foundation of inference based on those Discrete Random VariablesWe begin with a pair of discrete random variablesXandYand define thejoint (probability) massfunctionfX,Y(x,y) =P{X=x,Y=y}.Example having finite range, we can display the mass function in a 0 0 0 0 withunivariaterandom variables, we compute probabilities by adding the appropriate entries in {(X,Y) A}= (x,y) Af(X,Y)(x,y).Exercise {X=Y} {X+Y 3}. {XY= 0}. {X= 3}.
2 As before, the mass function has two basic properties. fX,Y(x,y) 0 for x,yfX,Y(x,y) = distribution of an individual random variable is call themarginal distribution . Themarginalmass functionforXis found by summing over the appropriate column and the Marginal mass functionforYcan be found be summing over the appropriate (x) = yfX,Y(x,y),fY(y) = xfX,Y(x,y)The Marginal mass functions for the example above arexfX(x) (y) two pairs of random variables with different Joint mass functions but the same marginalmass definition of expectation in the case of a finite sample spaceSis a straightforward generalization ofthe univarate (X,Y) = s Sg(X(s),Y(s))P{s}.From this formula, we see that expectation is again a positive linear functional. Using the distributiveproperty, we have the formulaEg(X,Y) = x,yg(x,y)fX,Y(x,y).
3 Exercise the example Continuous Random VariablesFor continuous random variables, we have the notion of thejoint (probability) density functionfX,Y(x,y) x y P{x<X x+ x,y<Y y+ y}.We can write this in integral form asP{(X,Y) A}= AfX,Y(x,y) basic properties of the Joint density function are fX,Y(x,y) 0 for ,Y(x,y)Figure 1: Graph of densityfX,Y(x,y) = 4(xy+x+y)/5, 0 x,y 1 fX,Y(x,y)dydx= (X,Y)have Joint densityfX,Y(x,y) ={c(xy+x+y)for 0 x 1,0 y 1, fX,Y(x,y)dydx= 10 10c(xy+x+y)dydx=c 10(12xy2+xy+12y2) 10dx=c 10(32x+12)dx=c(34x2+12x) 10=5c4andc= 4/5P{X Y}= 10 x045(xy+x+y)dydx=45 10(12xy2+xy+12y2) x0dx=45 10(12x3+32x2)dx=45(18x4+12x3) 10=45 58= cumulative distribution functionis defined asFX,Y(x,y) =P{X x,Y y}.For the case of continuous random variables, we haveFX,Y(x,y) = y x fX,Y(s,t) two applications of the fundamental theorem of calculus, we find that yFX,Y(x,y) = x fX,Y(s,y)dtand 2 x yFX,Y(x,y) =fX,Y(x,y).}
4 Example the density introduced above,FX,Y(x,y) = y0 x045(st+s+t)dtds= y045(12st2+st+12t2) y0ds= y045(12sy2+sy+12y2)ds=45(14s2y2+12s2y+12 sy2) x0=45(14x2y2+12x2y+12xy2)Notice thatFX,Y(1,1) = 1P{X 12,Y 12}=FX,Y(12,12)=45(14 14 14+12 14 12+12 12 14)=45 964= Joint cumulative distribution function is right continuous in each variable. It has limits at and+ similar to the univariate cumulative distribution function. limy FX,Y(x,y) = 0 and limx FX,Y(x,y) = 0. limx,y FX,Y(x,y) = addition,limy FX,Y(x,y) =FX(x) and limx FX,Y(x,y) =FY(y).Thus,FX(x) = x fX,Y(s,t)dtdsandFY(y) = y fX,Y(s,t) use the fundamental theorem of calculus to obtain themarginal (x) =F X(x) = fX,Y(x,t)dtandfY(y) =F Y(y) = fX,Y(s,y) the example density above, the Marginal densitiesfX(x) = 1045(xt+x+t)dt=45(12xt2+xt+12t2) 10=45(32x+12)andfY(y) =45(32y+12).
5 The formula for expectation for jointly continuous random variables is dervied by discretizingXandY,creating a double Rieman sum and taking a limit. This yields the identityEg(X,Y) = g(x,y)fX,Y(x,y) the example in the one-dimensional case, we can give a comprehensive formula for expectation using Riemann-Steiltjes integralsEg(X,Y) = g(x,y)dFX,Y(x,y).These can be realized as the limit of Riemann-Steiltjes sumsS(g,F) =m i=1n j=1g(xi,yj) FX,Y(xi,yj).Here, F(xi,yj) =P{xi<X xi+ x,yj<Y yj+ y}Exercise thatP{xi<X xi+ x,yj<Y yj}=FX,Y(xi+ x,yj+ y) FX,Y(xi,yj+ y) FX,Y(xi+ x,yj) +FX,Y(xi+ x,yj+ y).5
