As in the preceding example, let Z, V, and W be independent random variables with mean 0 and variance 1, and let X = Z + V and Y = Z + W. We have found that rho(X, Y) = 1/2. a) It follows that:
rho(X,-Y) =
p(-X,-Y) =
b) Suppose that X and Y are measured in dollars. Let X' and Y' be the same random variables, but measured in cents, so that X' = 100X and Y' = 100Y. Then,
p(X', Y') =
c) Suppose now that X = 3Z + 3V + 3 and Y -2Z-2 W. Then p(X, Y)
d) Suppose now that the variance of Z is replaced by a very large number. Then a(X, Y) is close to
e) Alternatively, suppose that the variance of Z is close to zero. Then (X, Y) is close to