"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.

f(x) = k/(x^7) ; x > 1

0 ; x<1
a. Determine the value of K for which f(x) is a legitimate pdf
b. Obtain the cumulative distribution function

F(x) = [ Blank ] ; x > 1

0 ; x < 1

c. Use the cdf from (b) to determine the probability that headway exceeds 2 sex. (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. (Roun to nearest 3 decimal places)
e. What is the probability that headway is within 1 standard deviation of the mean value?