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- "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. x >1 f(x) =10 xs1 (a) Determine the value of k for which f(x) is a legitimate pdf. (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.)2. Claim events on a portfolio of insurance policies follow a Poisson process with parameter A. Individual claim amounts follow a distribution X with density: f(x)=0.0122re001, g>0. The insurance company calculates premiums using a premium loading of 45%. (a) Derive the moment generating function Mx(t).I need help with parts “c”, “d, and “e” thanks.
- !"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. The pdf of X is the following. 0.17e-0.17(x - 0.5) - {0.17e-0.1 x ≥ 0.5 otherwise (a) What is the probability that time headway is at most 6 sec? (Round your answer to three decimal places.) f(x) = (b) What is the probability that time headway is more than 6 sec? At least 6 sec? (Round your answers to three decimal places.) more than 6 sec at least 6 sec (c) What is the probability that time headway is between 5 and 6 sec? (Round your answer to three decimal places.)Let X be a Poisson distribution with A as parameter: a. Find E(X) and VAR(X) b. Find the moment-generating function
- "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 o heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following form. x! f(x) = (a) Determine the value of k for which f(x) is a legitimate pdf. (b) Obtain the cumulative distribution function. F(x) = x< 1 (c) Use the cdf from (b) to determine the probability that headway exceeds 2 sec. (Round your answer to three decimal places.) 016 Use the cdf from (b) to determine the probability that headway is between 2 and sec. (Round your answer to three decimal places.) .014 (d) Obtain the mean value of headway and the standard deviation of headway. (Round your answers to three decimal places.) mean .6 standard deviation (e) What is the probability that headway is within 1…6. (b)Suppose that X has moment generating function Mx(t) = ( et +) (i) Find the p.m.f. of X. (ii)Find the mean and variance of X.If the pdf of a random variable X is f(x) = { 0.15-0.15(x-0.5), x ≥ 0.5 otherwise Find the moment generating function and use it to find the mean and variance of X.
- Ay 1Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤ TIn commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with A = 0 and B = 10, then it can be shown that the total waiting time Y has the pdf f (y) = = 100 Y 5 10 0 ≤ y ≤ 10 10 ≤ y ≤ 20 a. Sketch a graph of the pdf of Y (do on your own paper, you will not be turning this in. It is for your reference only.) [ f(y)dy=[ -∞ b. Verify that [ f (y) dy = 1 10 120 10 = 1 2 + c. What is the probability that the total waiting time is at most 8 minutes? d. What is the probability that the total waiting time is at most 13 minutes? = 1 e. What is the probability that the total waiting time is between 8 and 13 minutes? f. What is the probability that the total waiting time is either less than 7 minutes or more than 14 minutes?
