Question

(10 pts) Suppose that when I drive to school, I encounter one traffic light on Lewis Road and one traffic light on Santa Rosa Rd. Let the random variable X = number of red lights that I encounter on Lewis and Y = number of red lights that I encounter on Santa Rosa. Suppose that the marginal distributions of X and Y are as shown in the following probability table:

X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0
Notice that E(X) = E(Y) = .5, and Var(X) = Var(Y) = .25.

a) Fill in the table in such a way that Corr(X,Y) = 1. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0


b) Fill in the table in such a way that Corr(X,Y) = -1. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0


c) Fill in the table in such a way that Corr(X,Y) = 0. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0

Consider the variable W=X+Y, representing the total number of red lights I encounter on my drive to school.
d) Calculate E(W)





e) For each of the cases in parts a), b) and c), calculate SD(W)

Answer 1