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Unformatted text preview: Bivariate Distributions (Section 5.2) Often we are interested in the outcomes of 2 (or more) random variables. In the case of two random variables, we will label them X and Y. Suppose you have the opporunity to purchase shares of two firms. Your (subjective) joint probability distribution (p(x,y)) for the return on the two stocks is given below, where: p(x,y) = Prob(X=x and Y=y) (this is like an intersection of events in Chapter 6): Stock B Return (Y) Stock A Return (X) 0% 10%-5% 0.15 0.35 15% 0.35 0.15 For instance, the probability they both perform poorly (X=-5 and Y=0) is small (0.15). Also, the probaility that they both perform strongly (X=15 and Y=10) is small (0.15). Its more likely that one will perform strongly, while the other will perform weakly (X=15 and Y=0) or (X=-5 and Y=10), each outcome with probability 0.35. We can think of these firms as substitutes. Marginal Distributions Marginally, what is the probability distribution for stock A (this is called the marginal distribution)? For stock B? These are given in the following table, and are computed by summing the joint probabilities across the level of the other variable. Stock A Stock B x p(x)=p(x,0)+p(x,10) y p(y)=p(-5,y)+p(15,y) -5 .15+.35 = .50 0 .15+.35 = .50 15 .35+.15 = .50 10 .35+.15 = .50 Hence, we can compute the mean and variance for X and Y: = x all y all y x p y x all for y x p 1 ) , ( , 1 ) , ( So, both stocks have the same expected return, but stock A is riskier, in the sense that its variance is much larger. Note that the standard deviations are the square roots of the its variance is much larger....
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- Spring '08