Determine the value of k for which f(x) is a legitimate pdf. Obtain the cumulative distribution function. x > 1 F(x) xs1 O Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to four decimal places.) Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec. (Round your answer to four decimal places.)

MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
icon
Related questions
Question
**Time Headway in Traffic Flow Analysis**

"Time headway" in traffic flow refers to the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let \( X \) be the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (in seconds). Suppose that in a particular traffic environment, the distribution of time headway has the following form:

\[ 
f(x) = \begin{cases} 
\frac{k}{x^6} & x > 1 \\ 
0 & x \leq 1 
\end{cases} 
\]

**Tasks**

(a) Determine the value of \( k \) for which \( f(x) \) is a legitimate probability density function (pdf).

\[ \text{Answer: } \_\_\_\_\_\_ \]

(b) Obtain the cumulative distribution function (CDF).

\[ 
F(x) = \begin{cases} 
\quad\quad\quad \_ \quad\quad\quad & x > 1 \\ 
0 & x \leq 1 
\end{cases} 
\]

**Probability Calculations Using the CDF**

(c) Use the CDF to determine the probability that the headway exceeds 2 seconds. (Round your answer to four decimal places.)

\[ \text{Answer: } \_\_\_\_\_\_ \]

Use the CDF to determine the probability that headway is between 2 and 3 seconds. (Round your answer to four decimal places.)

\[ \text{Answer: } \_\_\_\_\_\_ \]

(d) Obtain the mean value of headway and the standard deviation of headway. (Round your standard deviation to three decimal places.)

\[ \text{Mean: } \_\_\_\_\_\_ \]

\[ \text{Standard Deviation: } \_\_\_\_\_\_ \]

(e) What is the probability that headway is within 1 standard deviation of the mean value? (Round your answer to three decimal places.)

\[ \text{Answer: } \_\_\_\_\_\_ \]

**Steps to Consider**

1. **Normalization:** Ensure \( f(x) \) integrates to 1 over its domain for it to be a valid probability density function.
2. **CDF Calculation:** Integrate \( f(x) \) to obtain \( F
Transcribed Image Text:**Time Headway in Traffic Flow Analysis** "Time headway" in traffic flow refers to the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let \( X \) be the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (in seconds). Suppose that in a particular traffic environment, the distribution of time headway has the following form: \[ f(x) = \begin{cases} \frac{k}{x^6} & x > 1 \\ 0 & x \leq 1 \end{cases} \] **Tasks** (a) Determine the value of \( k \) for which \( f(x) \) is a legitimate probability density function (pdf). \[ \text{Answer: } \_\_\_\_\_\_ \] (b) Obtain the cumulative distribution function (CDF). \[ F(x) = \begin{cases} \quad\quad\quad \_ \quad\quad\quad & x > 1 \\ 0 & x \leq 1 \end{cases} \] **Probability Calculations Using the CDF** (c) Use the CDF to determine the probability that the headway exceeds 2 seconds. (Round your answer to four decimal places.) \[ \text{Answer: } \_\_\_\_\_\_ \] Use the CDF to determine the probability that headway is between 2 and 3 seconds. (Round your answer to four decimal places.) \[ \text{Answer: } \_\_\_\_\_\_ \] (d) Obtain the mean value of headway and the standard deviation of headway. (Round your standard deviation to three decimal places.) \[ \text{Mean: } \_\_\_\_\_\_ \] \[ \text{Standard Deviation: } \_\_\_\_\_\_ \] (e) What is the probability that headway is within 1 standard deviation of the mean value? (Round your answer to three decimal places.) \[ \text{Answer: } \_\_\_\_\_\_ \] **Steps to Consider** 1. **Normalization:** Ensure \( f(x) \) integrates to 1 over its domain for it to be a valid probability density function. 2. **CDF Calculation:** Integrate \( f(x) \) to obtain \( F
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 3 images

Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
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 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…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
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
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman