A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly selected couple dining at this restaurant, let X = the cost of the man’s dinner and Y = the cost of the woman’s dinner. The joint pmf of X and Y is given in the following table:
p(x, y) | 12 | y 15 | 20 | |
12 | .05 | .05 | .10 | |
X | 15 | .05 | .10 | .35 |
20 | 0 | .20 | .10 |
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 $15 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. Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find the message “You will receive as a refund the difference between the cost of the more expensive and the less expensive meal that you have chosen.” How much would the restaurant expect to refund?
a.
Find the marginal pmf’s of X and Y.
Answer to Problem 75SE
The marginal pmf of X is:
X | 12 | 15 | 20 |
0.20 | 0.50 | 0.30 |
The marginal pmf and Y is:
Y | 12 | 15 | 20 |
0.10 | 0.35 | 0.55 |
Explanation of Solution
Given info:
In a restaurant serving three fixed-price dinners, X is the cost of a man’s dinner and Y is the cost of a woman’s dinner, for a randomly selected couple. The joint pmf of X and Y are given. The possible values of both X and Y are 12, 15 and 20.
Calculation:
Marginal pmf:
The marginal probability mass function or marginal pmf for each possible value x of a random variable X, when the joint pmf of X and Y is
The marginal pmf of X for
This is the same as the row total for the row containing
Thus, the marginal probability for each value of X is the row total for the corresponding value of X.
The marginal pmf of Y for
This is the same as the column total for the column containing
Thus, the marginal probability for each value of Y is the column total for the corresponding value of Y.
Thus, the marginal pmf for X and Y are found similarly as follows:
y | |||||
12 | 15 | 20 | Total | ||
x | 12 | 0.05 | 0.05 | 0.10 | 0.20 |
15 | 0.05 | 0.10 | 0.35 | 0.50 | |
20 | 0 | 0.20 | 0.10 | 0.30 | |
Total | 0.10 | 0.35 | 0.55 | 1 |
Thus, the marginal pmf of X and Y are:
X | 12 | 15 | 20 |
0.20 | 0.50 | 0.30 | |
Y | 12 | 15 | 20 |
0.10 | 0.35 | 0.55 |
b.
Find the probability that each of the man’s and the woman’s dinner cost at most $15.
Answer to Problem 75SE
The probability that each of the man’s and the woman’s dinner cost at most $15 is 0.25.
Explanation of Solution
Calculation:
The probability that each of the man’s and the woman’s dinner cost at most $15 is
Now,
From the data,
Thus,
Hence, the value of
c.
Decide whether X and Y are independent.
Answer to Problem 75SE
The random variables X and Y are not independent.
Explanation of Solution
Calculation:
Independent random variables:
The pair of random variables, X and Y are independent, if, for every pair of values, x and y, taken by the random variables is such that
Consider the pair of values
From the marginal pmf table,
Thus,
But from the data,
Thus,
Hence, the random variables X and Y are not independent.
d.
Find the expected total cost of the dinner for the two people.
Answer to Problem 75SE
The expected total cost of the dinner for the two people is $33.35.
Explanation of Solution
Calculation:
The total cost of the dinner for the two people is
Expectation of sum of random variables:
The expectation of sum of 2 random variables X and Y, having joint pmf
The calculation for
The following table shows the necessary calculations:
24 | 0.05 | 1.2 | |
27 | 0.05 | 1.35 | |
32 | 0.1 | 3.2 | |
27 | 0.05 | 1.35 | |
30 | 0.1 | 3 | |
35 | 0.35 | 12.25 | |
32 | 0 | 0 | |
35 | 0.2 | 7 | |
40 | 0.1 | 4 | |
Hence, the expected total cost of the dinner for the two people is $33.35.
e.
Find the amount of money the restaurant would expect to refund, if in the fortune cookies at the end of the meal, a couple finds the message “You will receive as a refund the difference between the cost of the more expensive and the less expensive meal that you have chosen.”
Answer to Problem 75SE
The amount of money the restaurant would expect to refund under the given scenario is $3.85.
Explanation of Solution
Justification:
The difference between the cost of the more expensive and the less expensive meal is
Expectation of a function of random variables:
The expectation of a function,
Thus, the expectation of
The calculation for
The following table shows the necessary calculations:
0 | 0.05 | 0 | |
3 | 0.05 | 0.15 | |
8 | 0.1 | 0.8 | |
3 | 0.05 | 0.15 | |
0 | 0.1 | 0 | |
5 | 0.35 | 1.75 | |
8 | 0 | 0 | |
5 | 0.2 | 1 | |
0 | 0.1 | 0 | |
Hence, the expected total cost of the dinner for the two people is $3.85.
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Chapter 5 Solutions
Probability and Statistics for Engineering and the Sciences
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