(a) Determine the value of k for which f(x) is a legitimate pdf. 5 (b) Obtain the cumulative distribution function. F(x) 1-x 0 X> 1 x≤ 1 (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to four decimal places.) 0.0156 X Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec. (Round your answer to four decimal places.) 0.0271
(a) Determine the value of k for which f(x) is a legitimate pdf. 5 (b) Obtain the cumulative distribution function. F(x) 1-x 0 X> 1 x≤ 1 (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to four decimal places.) 0.0156 X Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec. (Round your answer to four decimal places.) 0.0271
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![### Traffic Flow and Time Headway Distribution
**Concept:**
"Time headway" in traffic flow is 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 \( X \) represent the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (in seconds). In a particular traffic environment, the distribution of time headway is given by:
\[
f(x) =
\begin{cases}
\frac{k}{x^5} & \text{for } x > 1 \\
0 & \text{for } x \leq 1
\end{cases}
\]
**Tasks and Solutions:**
- **(a) Legitimate pdf Value of \( k \):**
Determine the value of \( k \) for which \( f(x) \) is a legitimate probability density function (pdf).
**Solution:** \( k = 5 \) ✔
- **(b) Cumulative Distribution Function:**
Calculate the cumulative distribution function \( F(x) \).
\[
F(x) =
\begin{cases}
1 - x^{-5} & \text{for } x > 1 \\
0 & \text{for } x \leq 1
\end{cases}
\]
**Solution:** Correct formulation ✔
- **(c) Probabilities Using CDF:**
1. Use the CDF from (b) to determine the probability that the headway exceeds 2 seconds. (Round your answer to four decimal places.)
**Incorrect Solution:** 0.0156 ✗
2. Use the CDF from (b) to determine the probability that the headway is between 2 and 3 seconds. (Round your answer to four decimal places.)
**Correct Solution:** 0.0271 ✔
- **(d) Mean and Standard Deviation:**
Calculate the mean value of headway and the standard deviation of headway. (Round the standard deviation to three decimal places.)
Fields to be filled:
- Mean
- Standard Deviation
- **(e) Probability Within One Standard Deviation:**
What is the probability that headway is within one standard deviation of the mean value? (Round your answer to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F390df6bc-8765-4adb-882a-689888791ad2%2F867ef191-6070-41d4-a53d-a1539f32aa5f%2F0nbqun8_processed.png&w=3840&q=75)
Transcribed Image Text:### Traffic Flow and Time Headway Distribution
**Concept:**
"Time headway" in traffic flow is 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 \( X \) represent the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (in seconds). In a particular traffic environment, the distribution of time headway is given by:
\[
f(x) =
\begin{cases}
\frac{k}{x^5} & \text{for } x > 1 \\
0 & \text{for } x \leq 1
\end{cases}
\]
**Tasks and Solutions:**
- **(a) Legitimate pdf Value of \( k \):**
Determine the value of \( k \) for which \( f(x) \) is a legitimate probability density function (pdf).
**Solution:** \( k = 5 \) ✔
- **(b) Cumulative Distribution Function:**
Calculate the cumulative distribution function \( F(x) \).
\[
F(x) =
\begin{cases}
1 - x^{-5} & \text{for } x > 1 \\
0 & \text{for } x \leq 1
\end{cases}
\]
**Solution:** Correct formulation ✔
- **(c) Probabilities Using CDF:**
1. Use the CDF from (b) to determine the probability that the headway exceeds 2 seconds. (Round your answer to four decimal places.)
**Incorrect Solution:** 0.0156 ✗
2. Use the CDF from (b) to determine the probability that the headway is between 2 and 3 seconds. (Round your answer to four decimal places.)
**Correct Solution:** 0.0271 ✔
- **(d) Mean and Standard Deviation:**
Calculate the mean value of headway and the standard deviation of headway. (Round the standard deviation to three decimal places.)
Fields to be filled:
- Mean
- Standard Deviation
- **(e) Probability Within One Standard Deviation:**
What is the probability that headway is within one standard deviation of the mean value? (Round your answer to
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