Solution Found!
Answer: Example 4.5 introduced the concept of time headway
Chapter 4, Problem 13E(choose chapter or problem)
Example \(4.5\) introduced the concept of time headway in traffic flow and proposed a particular distribution for \(X=\) the headway between two randomly selected consecutive cars (sec). Suppose that in a different traffic environment, the distribution of time headway has the form
\(f(x)=\left\{\begin{array}{cc}\frac{k}{x^{4}} & x>1 \\0 & x \leq 1\end{array}\right.\)
a. Determine the value of \(k\) for which \(f(x)\) is a legitimate pdf.
b. Obtain the cumulative distribution function.
c. Use the cdf from (b) to determine the probability that headway exceeds \(2 \mathrm{sec}\) and also the probability that headway is between 2 and \(3 \mathrm{sec}\).
d. Obtain the mean value of headway and the standard deviation of headway.
e. What is the probability that headway is within 1 standard deviation of the mean value?
Questions & Answers
QUESTION:
Example \(4.5\) introduced the concept of time headway in traffic flow and proposed a particular distribution for \(X=\) the headway between two randomly selected consecutive cars (sec). Suppose that in a different traffic environment, the distribution of time headway has the form
\(f(x)=\left\{\begin{array}{cc}\frac{k}{x^{4}} & x>1 \\0 & x \leq 1\end{array}\right.\)
a. Determine the value of \(k\) for which \(f(x)\) is a legitimate pdf.
b. Obtain the cumulative distribution function.
c. Use the cdf from (b) to determine the probability that headway exceeds \(2 \mathrm{sec}\) and also the probability that headway is between 2 and \(3 \mathrm{sec}\).
d. Obtain the mean value of headway and the standard deviation of headway.
e. What is the probability that headway is within 1 standard deviation of the mean value?
ANSWER:
Step 1 of 6
Given the distribution
\(f(x)=\begin{cases} \frac{k}{x^4}, & x>1 \\ 0, & x\leq 1 \end{cases}\)
Here, x = the headway between two randomly selected consecutive cars (sec).