1. A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly selected couple dinning at this restaurant, let X be the cost of the man's dinner and Y be the cost of the woman's dinner. The joint probability distribution is given below. Find the covariance Cov(X, Y). y f(x, y) $12 $15 $20 $12 0.05 0.05 0.10 $15 0.05 0.10 0.35 $20 0 0.20 0.10 X
1. A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly selected couple dinning at this restaurant, let X be the cost of the man's dinner and Y be the cost of the woman's dinner. The joint probability distribution is given below. Find the covariance Cov(X, Y). y f(x, y) $12 $15 $20 $12 0.05 0.05 0.10 $15 0.05 0.10 0.35 $20 0 0.20 0.10 X
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Can someone please help me to answer the following question
![### Problem Description
A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly selected couple dining at this restaurant, let \( X \) be the cost of the man's dinner and \( Y \) be the cost of the woman’s dinner. The joint probability distribution is given below. Find the covariance \( \text{Cov}(X, Y) \).
### Joint Probability Distribution
The table below represents the joint probability distribution \( f(x, y) \) of \( X \) and \( Y \):
| | \( y \) = $12 | \( y \) = $15 | \( y \) = $20 |
|----|:-------------:|:-------------:|:-------------:|
| \( x \) = $12 | 0.05 | 0.05 | 0.10 |
| \( x \) = $15 | 0.05 | 0.10 | 0.35 |
| \( x \) = $20 | 0 | 0.20 | 0.10 |
### Objective
Find the covariance \( \text{Cov}(X, Y) \).
### Explanation of the Table
1. The rows of the table correspond to the price of the man's dinner (\( x \)).
2. The columns of the table correspond to the price of the woman’s dinner (\( y \)).
3. Each cell in the table represents the joint probability \( f(x, y) \) for a given pair of dinner prices.
### Steps to Find Covariance
1. **Calculate the Expected Values \( E(X) \) and \( E(Y) \):**
\[
E(X) = \sum_{x} x \cdot P(X=x)
\]
\[
E(Y) = \sum_{y} y \cdot P(Y=y)
\]
2. **Calculate the Expected Value of the Product \( E(XY) \):**
\[
E(XY) = \sum_{x} \sum_{y} x \cdot y \cdot f(x, y)
\]
3. **Compute the Covariance \( \text{Cov}(X, Y) \):**
\[
\text{Cov}(X, Y) = E(XY) - E(X)E](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F322ab9b5-8b73-42ca-a79a-5c7ee3611144%2F99c4144a-a450-4569-b68f-ae067b53f599%2Fzu31fkr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Description
A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly selected couple dining at this restaurant, let \( X \) be the cost of the man's dinner and \( Y \) be the cost of the woman’s dinner. The joint probability distribution is given below. Find the covariance \( \text{Cov}(X, Y) \).
### Joint Probability Distribution
The table below represents the joint probability distribution \( f(x, y) \) of \( X \) and \( Y \):
| | \( y \) = $12 | \( y \) = $15 | \( y \) = $20 |
|----|:-------------:|:-------------:|:-------------:|
| \( x \) = $12 | 0.05 | 0.05 | 0.10 |
| \( x \) = $15 | 0.05 | 0.10 | 0.35 |
| \( x \) = $20 | 0 | 0.20 | 0.10 |
### Objective
Find the covariance \( \text{Cov}(X, Y) \).
### Explanation of the Table
1. The rows of the table correspond to the price of the man's dinner (\( x \)).
2. The columns of the table correspond to the price of the woman’s dinner (\( y \)).
3. Each cell in the table represents the joint probability \( f(x, y) \) for a given pair of dinner prices.
### Steps to Find Covariance
1. **Calculate the Expected Values \( E(X) \) and \( E(Y) \):**
\[
E(X) = \sum_{x} x \cdot P(X=x)
\]
\[
E(Y) = \sum_{y} y \cdot P(Y=y)
\]
2. **Calculate the Expected Value of the Product \( E(XY) \):**
\[
E(XY) = \sum_{x} \sum_{y} x \cdot y \cdot f(x, y)
\]
3. **Compute the Covariance \( \text{Cov}(X, Y) \):**
\[
\text{Cov}(X, Y) = E(XY) - E(X)E
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