In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the walting time (in minutes) at each stop has a uniform distribution with A = 0 and B = 5, can be shown that the total waiting time Y has the pdf below. Osy<5 f(y) -{ 2 5sys 10 y <0 or y > 10 (a) Sketch a graph of the pdf of Y. f(y) f(y) f(y) 0.20 0.20- 0.20 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 0.05 y 10 y 10 4. 6 8. 10 4. 8. 4. f(y) 0.20 0.15 0.10 0.05 y 10
In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the walting time (in minutes) at each stop has a uniform distribution with A = 0 and B = 5, can be shown that the total waiting time Y has the pdf below. Osy<5 f(y) -{ 2 5sys 10 y <0 or y > 10 (a) Sketch a graph of the pdf of Y. f(y) f(y) f(y) 0.20 0.20- 0.20 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 0.05 y 10 y 10 4. 6 8. 10 4. 8. 4. f(y) 0.20 0.15 0.10 0.05 y 10
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
![In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with \(A = 0\) and \(B = 5\), then it can be shown that the total waiting time \(Y\) has the probability density function (pdf) below:
\[
f(y) =
\begin{cases}
\frac{1}{25}y & 0 \leq y < 5 \\
\frac{2}{5} - \frac{1}{25}y & 5 \leq y \leq 10 \\
0 & y < 0 \text{ or } y > 10
\end{cases}
\]
(a) Sketch a graph of the pdf of \(Y\).
### Graph Explanations
1. **Plot 1**:
- Represents a linear increase in the pdf from \(y = 0\) to \(y = 5\).
- The line starts at the origin (0,0) and rises linearly to \( (5, 0.20) \).
2. **Plot 2**:
- Displays a triangular shape with the peak at \(y = 5\).
- Increases linearly from (0,0) to peak at \((5,0.20)\), then decreases to \((10,0)\).
3. **Plot 3**:
- Shows an inverted linear increase from \(y = 0\) to \(y = 5\).
- The line declines from \( (0, 0.20) \) to \( (5, 0) \), then remains zero till \(y = 10\).
4. **Plot 4**:
- Illustrates a linear increase from \((5,0)\) to \((10, 0.20)\).
- The line increases and forms a right triangular pattern.
From the given graphs, the correct representation is Plot 2, which accurately matches the piecewise function description—a triangle with its peak at \(y = 5\) and base spanning from \(0\) to \(10\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4fac401-f688-4a8e-b637-a2500ce6ac46%2F56ebc552-8252-41b5-bba5-2d5f18d09396%2Fjan9kw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with \(A = 0\) and \(B = 5\), then it can be shown that the total waiting time \(Y\) has the probability density function (pdf) below:
\[
f(y) =
\begin{cases}
\frac{1}{25}y & 0 \leq y < 5 \\
\frac{2}{5} - \frac{1}{25}y & 5 \leq y \leq 10 \\
0 & y < 0 \text{ or } y > 10
\end{cases}
\]
(a) Sketch a graph of the pdf of \(Y\).
### Graph Explanations
1. **Plot 1**:
- Represents a linear increase in the pdf from \(y = 0\) to \(y = 5\).
- The line starts at the origin (0,0) and rises linearly to \( (5, 0.20) \).
2. **Plot 2**:
- Displays a triangular shape with the peak at \(y = 5\).
- Increases linearly from (0,0) to peak at \((5,0.20)\), then decreases to \((10,0)\).
3. **Plot 3**:
- Shows an inverted linear increase from \(y = 0\) to \(y = 5\).
- The line declines from \( (0, 0.20) \) to \( (5, 0) \), then remains zero till \(y = 10\).
4. **Plot 4**:
- Illustrates a linear increase from \((5,0)\) to \((10, 0.20)\).
- The line increases and forms a right triangular pattern.
From the given graphs, the correct representation is Plot 2, which accurately matches the piecewise function description—a triangle with its peak at \(y = 5\) and base spanning from \(0\) to \(10\).
![(b) Verify that \(\int_{-\infty}^{\infty} f(y) \, dy = 1\).
\[
\int_{-\infty}^{\infty} f(y) \, dy = \left[ \boxed{} \right]_0^5 + \left[ \boxed{} \right]_5^{10}
\]
\[
= \frac{1}{2} + \left( \boxed{} \right)
\]
\[
= \boxed{}
\]
(c) What is the probability that total waiting time is at most 4 min?
\[
\boxed{0.32} \checkmark
\]
(d) What is the probability that total waiting time is at most 9 min?
\[
\boxed{}
\]
(e) What is the probability that total waiting time is between 4 and 9 min?
\[
\boxed{}
\]
(f) What is the probability that total waiting time is either less than 2 min or more than 6 min?
\[
\boxed{}
\]
- Need Help? [Read It]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4fac401-f688-4a8e-b637-a2500ce6ac46%2F56ebc552-8252-41b5-bba5-2d5f18d09396%2Fkn103t7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) Verify that \(\int_{-\infty}^{\infty} f(y) \, dy = 1\).
\[
\int_{-\infty}^{\infty} f(y) \, dy = \left[ \boxed{} \right]_0^5 + \left[ \boxed{} \right]_5^{10}
\]
\[
= \frac{1}{2} + \left( \boxed{} \right)
\]
\[
= \boxed{}
\]
(c) What is the probability that total waiting time is at most 4 min?
\[
\boxed{0.32} \checkmark
\]
(d) What is the probability that total waiting time is at most 9 min?
\[
\boxed{}
\]
(e) What is the probability that total waiting time is between 4 and 9 min?
\[
\boxed{}
\]
(f) What is the probability that total waiting time is either less than 2 min or more than 6 min?
\[
\boxed{}
\]
- Need Help? [Read It]
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 2 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