Transcription of 6 Jointly continuous random variables
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6 Jointly continuous random variables
6 Jointly continuous random variablesAgain, we deviate from the order in the book for this chapter, so the subsec-tions in this chapter do not correspond to those in the Joint density functionsRecall thatXis continuous if there is a functionf(x) (the density) such thatP(X t) =Zt fX(x)dxWe generalize this to two random random variablesXandYare Jointly continuous if thereis a functionfX,Y(x, y)onR2, called the joint probability density function,such thatP(X s, Y t) =Z Zx s,y tfX,Y(x, y)dxdyThe integral is over{(x, y) :x s, y t}. We can also write the integral asP(X s, Y t) =Zs Zt fX,Y(x, y)dy dx=Zt Zs fX,Y(x, y)dx dyIn order for a functionf(x, y) to be a joint density it must satisfyf(x, y) 0Z Z f(x, y)dxdy= 1 Just as with one random variable, the joint density functioncontains allthe information about the underlying probability measure if we only look atthe random variablesXandY. In particular, we can compute the probabilityof any event defined in terms ofXandYjust usingf(x, y).
6 Jointly continuous random variables Again, we deviate from the order in the book for this chapter, so the subsec-tions in this chapter do not correspond to those in the text. 6.1 Joint density functions Recall that X is continuous if there is a function f(x) (the density) such that P(X ≤ t) = Z t −∞ fX(x)dx
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