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.
Recall that X is continuous if there is a function f(x) (the density) such that P(X ≤ t) = Z t −∞ fX(x)dx We generalize this to two random variables. Definition 1. Two random variables X and Y are jointly continuous if there is a function fX,Y (x,y) on R2, called the joint probability density function, such that P(X ≤ s,Y ≤ t) = Z Z ...
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