4.77 A researcher suspected that the number of betweenmeal snacks eaten by students in a day during final examinations might depend on the number of tests a student had to take on that day. The accompanying table shows joint probabilities, estimated from a survey.

Number of Snacks (Y)

Number of Tests (X)

0 1 2 3 0 0.07 0.09 0.06 0.01 1 0.07 0.06 0.07 0.01 2 0.06 0.07 0.14 0.03 3 0.02 0.04 0.16 0.04

a. Find the probability distribution of X and compute the mean number of tests taken by students on that day.

b. Find the probability distribution of Y and, hence, the mean number of snacks eaten by students on that day.

c. Find and interpret the conditional probability distribution of Y, given that X = 3.

d. Find the covariance between X and Y.

e. Are number of snacks and number of tests independent of each other?