"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 consecuti 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. x> 1 f(x) =10 (a) Determine the value of k for which f(x) is a legitimate pdf. 9 (b) Obtain the cumulative distribution function. x > 1 F(x) = xS1 (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 consecuti 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. x> 1 f(x) =10 (a) Determine the value of k for which f(x) is a legitimate pdf. 9 (b) Obtain the cumulative distribution function. x > 1 F(x) = xS1 (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.)
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
x > 1
f(x)
,10
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.)
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