91Ó°ÊÓ

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: \begin{tabular}{cc|ccc} \(p(x, y)\) & & 12 & 15 & 20 \\ \hline & 12 & \(.05\) & \(.05\) & \(.10\) \\ \multirow{x}{*}{} & 15 & \(.05\) & \(.10\) & \(.35\) \\ & 20 & 0 & \(.20\) & \(.10\) \end{tabular} 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?

Step by step solution

01

Compute the Marginal PMF of X

To find the marginal probability mass function (pmf) of X, we need to sum the joint probabilities over all possible values of Y. This involves summing each row of the table.- For \( X = 12 \): \( P(X=12) = 0.05 + 0.05 + 0.10 = 0.20 \)- For \( X = 15 \): \( P(X=15) = 0.05 + 0.10 + 0.35 = 0.50 \)- For \( X = 20 \): \( P(X=20) = 0 + 0.20 + 0.10 = 0.30 \)

02

Compute the Marginal PMF of Y

To find the marginal pmf of Y, we sum the joint probabilities over all possible values of X, summing each column of the table.- For \( Y = 12 \): \( P(Y=12) = 0.05 + 0.05 + 0 = 0.10 \)- For \( Y = 15 \): \( P(Y=15) = 0.05 + 0.10 + 0.20 = 0.35 \)- For \( Y = 20 \): \( P(Y=20) = 0.10 + 0.35 + 0.10 = 0.55 \)

03

Determine Probability of Meals Costing at Most $15 Each

The probability that both meals cost at most \$15 is the sum of the probabilities where this condition is true:- \(P(X=12, Y=12) = 0.05 \)- \(P(X=12, Y=15) = 0.05 \)- \(P(X=15, Y=12) = 0.05 \)- \(P(X=15, Y=15) = 0.10 \)So, the total probability is \( 0.05 + 0.05 + 0.05 + 0.10 = 0.25 \).

04

Check Independence of X and Y

X and Y are independent if and only if \( P(X = x, Y = y) = P(X = x) \times P(Y = y) \) for all x and y. We need to check this for each pair:- For \( (X=12, Y=12): \ 0.05 eq 0.20 \times 0.10 \) (product is 0.02).As one counter-example suffices, we conclude that X and Y are not independent.

05

Compute Expected Total Cost

The expected total cost of dinner is \( E(X + Y) = E(X) + E(Y) \). First, calculate the expected value of X and Y separately.\( E(X) = 12(0.20) + 15(0.50) + 20(0.30) = 2.4 + 7.5 + 6 = 15.9 \).\( E(Y) = 12(0.10) + 15(0.35) + 20(0.55) = 1.2 + 5.25 + 11 = 17.45 \).Thus, \( E(X + Y) = 15.9 + 17.45 = 33.35 \).

06

Compute Expected Refund

The expected refund is calculated by considering cases where one meal costs more than the other and weighing by the joint probability.- For each pair \((X, Y)\): compute \(|X - Y| \) and multiply by \(P(X, Y)\).Compute these for all pairs and sum them:- \( |12 - 15| \times 0.05 + |12 - 20| \times 0.10 + |15 - 12| \times 0.05 + ... \) - Effective computation leads to the values: 0.15 for \(X=12, Y=15\); 0.80 for \(X=12, Y=20\); 0.15 for \(X=15, Y=12\), and additional contributions, resulting in a total expected refund value of \(2.45 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Marginal Probabilities

When dealing with joint probability mass functions (PMFs), one important concept is marginal probabilities. These are the probabilities associated with one variable, regardless of what the other variable might be. To find marginal probabilities associated with a variable, we sum the joint probabilities across all possible values of the other variable. This focuses only on how often each individual variable might appear.

For variable \( X \), which represents the cost of the man's dinner, the marginal probabilities are found by summing across all possible values of \( Y \), the woman's dinner cost. For example, the probability that \( X = 12 \) is calculated by adding the probabilities of all scenarios where the man's dinner costs \( \$12 \):

• \( P(X=12) = 0.05 + 0.05 + 0.10 = 0.20 \)

Similar calculations apply for \( X = 15 \) and \( X = 20 \).

Using the same method for \( Y \), summing across all \( X \) values gives us the marginal PMF for \( Y \):

• \( P(Y=12) = 0.05 + 0.05 + 0 = 0.10 \)
• \( P(Y=15) = 0.05 + 0.10 + 0.20 = 0.35 \)
• \( P(Y=20) = 0.10 + 0.35 + 0.10 = 0.55 \)

Recognizing these probabilities makes it easier to understand the behavior of each variable independently.

Independence of Random Variables

Two random variables are independent if knowing the value of one of them provides no information about the other. Mathematically, this means the joint probability of two variables equals the product of their individual marginal probabilities for all combinations of values. In other words, for random variables \( X \) and \( Y \), independence occurs when

• \( P(X=x, Y=y) = P(X=x) \cdot P(Y=y) \)

for all \( (x, y) \) pairs.

To check for independence in our restaurant example, take a pair like \( (X=12, Y=12) \). Calculating, we find:

• \( P(X=12, Y=12) = 0.05 \)
• \( P(X=12) \times P(Y=12) = 0.20 \times 0.10 = 0.02 \)

Since \( 0.05 eq 0.02 \), \( X \) and \( Y \) are not independent. Even a single counter-example such as this one indicates dependence between the variables. It's crucial to check multiple combinations to confirm any such relationship.

Expected Value Calculation

The expected value of a random variable provides a measure of its central tendency, essentially acting as a weighted average of all possible outcomes. When dealing with the total cost of dinners, you find the expected value for each variable separately and then sum them.

For the dinner costing scenario:

• Calculate \( E(X) \), the expected cost for the man's dinner:
  \( E(X) = 12(0.20) + 15(0.50) + 20(0.30) = 15.9 \)


• Calculate \( E(Y) \), the expected cost for the woman's dinner:
  \( E(Y) = 12(0.10) + 15(0.35) + 20(0.55) = 17.45 \)

Adding these values gives us \( E(X+Y) \), the expected total dinner cost:
\( 15.9 + 17.45 = 33.35 \).

This calculation highlights the average amount a couple might expect to spend. It's a valuable insight, especially when considering budgeting or analyzing spending habits over time.