A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation.

a. What is P(X = 1 and Y = 1)?

b. Compute P(X ≤ 1 and Y ≤ 1).

c. Give a word description of the event {X ≠0 and Y ≠0}, and compute the probability of this event.

d. Compute the marginal pmf of X and of Y. Using pX(x), what is P(X ≤ 1)?

e. Are X and Y independent rv's? Explain.

a) P=0.2

b) P=0.42

c) P=0.7

d) Marginal probability distribution of x:

Marginal probability distribution of y:

e) The two variables are dependant.

Step-by-step explanation:

The joint pmf of X and Y appears in the accompanying tabulation (attached).

a) Is the proability that one hose being used in the self service island and one hose being use in the full service island.

(according to the tabulation)

b)

c) This event means that at least one hose is in use in each of the island.

d)

Marginal probability distribution of x:

Marginal probability distribution of y:

Compute P(X≤1)

e) For X and Y to be independent variables, the following conditions must be true:

We evaluate the second probability for X=1 and Y=1.

This condition is not satisfied, so the two variables are dependant.