"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 = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following form. k f(x) = x > 1 x6 (a) Determine the value of k for which f(x) is a legitimate pdf. (b) Obtain the cumulative distribution function. x > 1 F(x) = XS 1 (c) 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.)
"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 = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following form. k f(x) = x > 1 x6 (a) Determine the value of k for which f(x) is a legitimate pdf. (b) Obtain the cumulative distribution function. x > 1 F(x) = XS 1 (c) 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.)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:"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 = the time headway for two randomly chosen consecutive cars on a freeway during a
period of heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following
form.
k
f(x) =
x > 1
x6
x< 1
(a) Determine the value of k for which f(x) is a legitimate pdf.
(b) Obtain the cumulative distribution function.
x > 1
F(x) =
x< 1
(c) 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.)
(d) Obtain the mean value of headway and the standard deviation of headway. (Round your standard deviation to three decimal
places.)
mean
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.)
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