Question

"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) = 6 10 X51 (a) Determine the value of k for which f(x) is a legitimate pdf. (b) Obtain the cumulative distribution function. F(x) = X>1 x51 VI (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to three decimal places.) Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec. (Round your answer to three decimal places.) (d) Obtain the mean value of headway and the standard deviation of headway. (Round your answers 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.)

Answer 1

f(x) = s k/6 NDI noi HCl a) Stix de Kdo al J > 6 - 6+) &k=15 b) Fla) = Ses parede = 5 [x - Etligt ali C) Plxsq) - Pixse) = 1 - Fly) bei 0.969 | d) P (2 2X(3) = F (3) - F (2) - 11-5s) - (-) - 10.027

(d) E(x) = gutxi da - Forside S + -5ti Mean Mean = {1-0] = 11.250] EG) - Ju xsd 5[x170m - 4 +1 -4+1 = 11.667 Standard deviation - ✓ E(x²) - F(x) Ja +10.323]

1.250+0.230 (e) p (1x-4126) = ls da X6 1.250-0.323 = 5TX abt19 1.573 BORA I [ 0.896 0.896