A restaurant serves three fixed price dinners costing 171925

A restaurant serves three fixed price dinners costing $17,$19,$25.
For a randomly selected man & woman pair dining at this restaurant,
let X = the cost of the man\'s dinner
let Y = the cost of the woman\'s dinner

The joint pmf of X and Y is given in the following table:

        |      x
        |
p(x,y) | 17 19 25
--------|--------------
     17 | .01 .07 .05
y    19 | .07 .03 .22
     25 | .05 .17 .33

(a) Compute the marginal pmf\'s of X and Y.

(b) What is the probability that the man\'s and the
    woman\'s dinner cost at most $19 each?

(c) Are X and Y independent? Justify Your Answer.

(d) What is the expected total cost of the dinner for the two people?

(e) Compute the covariance for X and Y.

(f) Suppose that when a pair opens a fortune cookie at the
    conclusion of the meal, they find the message --
    \"You will receive as refund the difference between the cost
     of the more expensive and the less expensive meal that you
     have chosen.\" How much does the restaurant expect to refund?

Solution

Let X and Y be two discrete random variables dened on the
sample space S of an experiment. The joint probability mass
function p(x; y) is dened for each pair of numbers (x; y) by
p(x; y) = P(X = x and Y = y)
(It must be the case that p(x; y) 0 and
P
x
P
y
p(x; y) = 1.)
For any event A consisting of pairs of (x; y), the probability
P[(X; Y ) 2 A] is obtained by summing the joint pmf over pairs in
A:
P[(X; Y ) 2 A] =
XX
(x;y)2A
p(x; y)