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.)
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
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4fac401-f688-4a8e-b637-a2500ce6ac46%2F22d31200-513c-42a2-b22e-2207f4e513ef%2Fn7srdr_processed.jpeg&w=3840&q=75)
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
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