An Instructor has given a short quiz consisting of two parts. For a randomly selected student let X = the number of points earned on the 1st part of the quiz and let Y = the number of points earned on the 2nd part. Suppose that the joint pmf of X and Y is given in the following table: _px(x,y) a. b. C. d. 0 X 5 10 0 0.02 0.04 0.01 Y 5 0.06 0.15 0.15 10 0.02 0.2 0.14 15 0.1 0.1 0.01 If the score recorded in the gradebook is the total number of points earned on the two parts, what is the expected recorded score E[X + Y]? Compute the covariance between X and Y. Compute the correlation coefficient, Pxy, for this X and Y. Does it appear that there is an association between the number of points earned on the two parts of this quiz? Explain your reasoning.
An Instructor has given a short quiz consisting of two parts. For a randomly selected student let X = the number of points earned on the 1st part of the quiz and let Y = the number of points earned on the 2nd part. Suppose that the joint pmf of X and Y is given in the following table: _px(x,y) a. b. C. d. 0 X 5 10 0 0.02 0.04 0.01 Y 5 0.06 0.15 0.15 10 0.02 0.2 0.14 15 0.1 0.1 0.01 If the score recorded in the gradebook is the total number of points earned on the two parts, what is the expected recorded score E[X + Y]? Compute the covariance between X and Y. Compute the correlation coefficient, Pxy, for this X and Y. Does it appear that there is an association between the number of points earned on the two parts of this quiz? Explain your reasoning.
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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![### Quiz Analysis using Joint PMF
#### Introduction
An instructor has given a short quiz consisting of two parts. For a randomly selected student, let \( X \) represent the number of points earned on the 1st part of the quiz and \( Y \) represent the number of points earned on the 2nd part. The joint probability mass function (pmf) of \( X \) and \( Y \) is provided in the following table:
#### Joint PMF Table
\[ p_{XY}(x, y) \]
| \( X \backslash Y \) | 0 | 5 | 10 | 15 |
|---|------|------|------|------|
| 0 | 0.02 | 0.06 | 0.02 | 0.1 |
| 5 | 0.04 | 0.15 | 0.2 | 0.1 |
| 10| 0.01 | 0.15 | 0.14 | 0.01 |
### Questions and Solutions
a. **Expected Total Recorded Score \( E[X + Y] \):**
- Compute the expected value of \( X + Y \) by calculating \( E[X] \) and \( E[Y] \), then summing them up.
b. **Covariance Between \( X \) and \( Y \):**
- Compute the covariance to understand how \( X \) and \( Y \) vary together.
c. **Correlation Coefficient \( \rho_{XY} \):**
- Calculate the correlation coefficient to measure the strength and direction of the linear relationship between \( X \) and \( Y \).
d. **Association Between \( X \) and \( Y \):**
- Analyze whether there is an association between the number of points earned on the two parts of the quiz and explain the reasoning.
### Detailed Steps for Solution:
**1. Expected Value of X and Y:**
\[
E[X] = \sum_{x} x \times P(X=x)
\]
Calculate the marginal probability for \( X \) and \( Y \) and compute their expected values.
**2. Expected Value of X + Y:**
\[
E[X+Y] = E[X] + E[Y]
\]
**3. Computing Covariance \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8952a85a-85f1-4c32-8b45-c088a1def42e%2F8b7ef83c-8d5e-4e68-9b07-5da3239a80e2%2Fovh5ocj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Quiz Analysis using Joint PMF
#### Introduction
An instructor has given a short quiz consisting of two parts. For a randomly selected student, let \( X \) represent the number of points earned on the 1st part of the quiz and \( Y \) represent the number of points earned on the 2nd part. The joint probability mass function (pmf) of \( X \) and \( Y \) is provided in the following table:
#### Joint PMF Table
\[ p_{XY}(x, y) \]
| \( X \backslash Y \) | 0 | 5 | 10 | 15 |
|---|------|------|------|------|
| 0 | 0.02 | 0.06 | 0.02 | 0.1 |
| 5 | 0.04 | 0.15 | 0.2 | 0.1 |
| 10| 0.01 | 0.15 | 0.14 | 0.01 |
### Questions and Solutions
a. **Expected Total Recorded Score \( E[X + Y] \):**
- Compute the expected value of \( X + Y \) by calculating \( E[X] \) and \( E[Y] \), then summing them up.
b. **Covariance Between \( X \) and \( Y \):**
- Compute the covariance to understand how \( X \) and \( Y \) vary together.
c. **Correlation Coefficient \( \rho_{XY} \):**
- Calculate the correlation coefficient to measure the strength and direction of the linear relationship between \( X \) and \( Y \).
d. **Association Between \( X \) and \( Y \):**
- Analyze whether there is an association between the number of points earned on the two parts of the quiz and explain the reasoning.
### Detailed Steps for Solution:
**1. Expected Value of X and Y:**
\[
E[X] = \sum_{x} x \times P(X=x)
\]
Calculate the marginal probability for \( X \) and \( Y \) and compute their expected values.
**2. Expected Value of X + Y:**
\[
E[X+Y] = E[X] + E[Y]
\]
**3. Computing Covariance \(
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