Example 4.5 introduced the concept of time headway in traffic flow and proposed a particular distribution for X = the headway between two randomly selected consecutive cars (sec). Suppose that in a different traffic environment, the distribution of time headway has the form a.? etermine the value of ?k ?for which ?f?(?x?) is a legitimate pdf. b.? ?Obtain the cumulative distribution function. c.? ?Use the cdf from (b) to determine the probability that headway exceeds 2 sec and also the probability that headway is between 2 and 3 sec. d.? ?Obtain the mean value of headway and the standard deviation of headway. e.? ?What is the probability that headway is within 1 standard deviation of the mean value?

Answer: Step1: Given the distribution of time headway has the form Here, X = the headway between two randomly selected consecutive cars (sec). Step2: a). We have to determine the value of k for which f(x) is a legitimate pdf. Given distribution is k f(x) = x4 Take Integration on both sides. f(x) dx = k4dx 1 x Integrate both sides with respect to ‘x’. k f(x) dx = x dx 1 Take the constants out Then, f(x) dx = k 14dx 1 x Here applying the power rule a xa+1 [ x dx = a+1 ] Here, f(x) dx = 1 for legitimacy. That is, x3 1 = k 3) 1 1 = k ( 1 - 0) 3 k 3 = 1 k = 3. Therefore, k = 3, for f(x s a legitimate pdf. Step3: b). The aim is to obtain the cumulative distribution function. The cumulative distribution function is x f(x) dx = k 4dx 1 x = k (x3)x 3 1 k 1 = 3 ( 1-( ))x3 Substitute k = 3. 1 = -1( 1-( ))x3 1 = (1- x3 Therefore, the cumulative distribution function is Step4: c). Use the cdf form (b), to determine the probability that headway exceeds 2 seconds. Therefore, P(x >2) = 1- P(x < 1) = 1- F(2) = 1- (1- 3) 2 = 1- (1-1 ) 1 8 8 1 Therefore, the probability that headway exceeds 2 seconds is 8 . Now, also the probability...