From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. Suppose X is the number of oranges and Y is the number of apples in the sample. (a) Find the joint probability distribution of X and Y. (b) Find P[(X,Y)EA], where A is the region that is given by {(x,y) | x+y<3}. C (a) Complete the joint probability distribution below. (Type integers or simplified fractions.) X f(x,y) 0 3 4 y 0 2
From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. Suppose X is the number of oranges and Y is the number of apples in the sample. (a) Find the joint probability distribution of X and Y. (b) Find P[(X,Y)EA], where A is the region that is given by {(x,y) | x+y<3}. C (a) Complete the joint probability distribution below. (Type integers or simplified fractions.) X f(x,y) 0 3 4 y 0 2
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Transcribed Image Text:### Joint Probability Distribution of Oranges and Apples in a Random Sample
#### Problem Statement
From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. Suppose \( X \) is the number of oranges and \( Y \) is the number of apples in the sample.
(a) Find the joint probability distribution of \( X \) and \( Y \).
(b) Find \( P((X,Y) \in A) \), where \( A \) is the region that is given by \(\{(x,y) | x + y \leq 3\}\).
#### Solution
(a) **Complete the joint probability distribution below.**
| \( f(x,y) \) | \( y \\ x \) | 0 | 1 | 2 | 3 | 4 |
|--------------|--------------|---|---|---|---|---|
| 0 | | | | | | |
(b) **Finding \( P((X,Y) \in A) \):**
To solve for the probability, one must integrate or sum over the joint distribution \( f(x,y) \) for all \((x,y)\) in the region \( A \).
![### Joint Density Function and Marginal Density
#### Introduction
A fast-food restaurant operates both a drive-through facility and a walk-in facility. On a randomly selected day, let \(X\) and \(Y\) represent the proportions of the time that the drive-through and walk-in facilities are in use, respectively. The joint density function of these random variables is given by:
\[
f(x, y) =
\begin{cases}
\frac{2}{11} (6x + 5y), & 0 \leq x \leq 1, \ 0 \leq y \leq 1 \\
0, & \text{elsewhere}
\end{cases}
\]
#### Problem Statement
Complete parts (a) through (c).
#### Part (a): Marginal Density of \(X\)
##### Task
Find the marginal density of \(X\).
##### Instructions
Select the correct choice below and fill in the answer box to complete your choice.
##### Choices
- \( A. \ h(y) = \Box, \ \text{for} \ 0 \leq y \leq 1 \)
- \( B. \ g(x) = \Box, \ \text{for} \ 0 \leq x \leq 1 \)
---
### Detailed Explanation
- To find the marginal density function \( g(x) \) of \(X\), you integrate the joint density \( f(x, y) \) over all possible values of \( y \).
\[ g(x) = \int_{0}^{1} f(x, y) \, dy \]
- Similarly, to find the marginal density function \( h(y) \) of \(Y\), you integrate the joint density \( f(x, y) \) over all possible values of \( x \).
\[ h(y) = \int_{0}^{1} f(x, y) \, dx \]
This problem challenges students to use their knowledge of joint density functions and marginal distributions to derive the marginal density of a given variable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7a11c6-35df-4daf-855c-043e8f69fb75%2F36aea7bf-6f26-4ac0-9289-287072dcfa57%2Fft3pw48_processed.png&w=3840&q=75)
Transcribed Image Text:### Joint Density Function and Marginal Density
#### Introduction
A fast-food restaurant operates both a drive-through facility and a walk-in facility. On a randomly selected day, let \(X\) and \(Y\) represent the proportions of the time that the drive-through and walk-in facilities are in use, respectively. The joint density function of these random variables is given by:
\[
f(x, y) =
\begin{cases}
\frac{2}{11} (6x + 5y), & 0 \leq x \leq 1, \ 0 \leq y \leq 1 \\
0, & \text{elsewhere}
\end{cases}
\]
#### Problem Statement
Complete parts (a) through (c).
#### Part (a): Marginal Density of \(X\)
##### Task
Find the marginal density of \(X\).
##### Instructions
Select the correct choice below and fill in the answer box to complete your choice.
##### Choices
- \( A. \ h(y) = \Box, \ \text{for} \ 0 \leq y \leq 1 \)
- \( B. \ g(x) = \Box, \ \text{for} \ 0 \leq x \leq 1 \)
---
### Detailed Explanation
- To find the marginal density function \( g(x) \) of \(X\), you integrate the joint density \( f(x, y) \) over all possible values of \( y \).
\[ g(x) = \int_{0}^{1} f(x, y) \, dy \]
- Similarly, to find the marginal density function \( h(y) \) of \(Y\), you integrate the joint density \( f(x, y) \) over all possible values of \( x \).
\[ h(y) = \int_{0}^{1} f(x, y) \, dx \]
This problem challenges students to use their knowledge of joint density functions and marginal distributions to derive the marginal density of a given variable.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman