Alana and Eva are sisters. Let X denote the number of birds that Alana may have in the next two years, and let Y denote the number of birds Eva may have during the same period. The marginal probability distributions of X and Y are shown below.

x 0 1 2 y 0 1 2

p(x) 0.5 0.3 0.2 p(y) 0.4 0.5 0.1

a. If X and Y are independent, find their bivariate distribution.

b. Compute the coefficient of correlation between X and Y. Did you expect this result? Why?

c. Determine the probability distribution of the random variable X + Y.

d. Calculate E(X + Y) and V(X + Y) directly by using the probability distribution of X + Y.