"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?

See detailed explanation.

Step-by-step explanation:

a) Recall that if a function is a pdf, then it must happen that . Then,

which gives as k = 6.

b). Recall that Then, F(x) =0 if . Then if x>1 we have from the previous point that

c). We want to know

The case

d). We have that . Where,

Then,

If k<6, then

So, the mean (k=1) is 6/(6-1) = 6/5= 1.2 . And the standard deviation is .

e. We are asked for Which is equivalent to

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system of linear equation

step-by-step explanation:

as given in the data

a= 20 , b= -12

let us suppose that it is a straight line equation

now we know the formula for the system of linear equation

b= ma + c )

here m is slope of a straight line which is equal to formula

m=   (ii)

and here c is a constant

now putting the value of a and b in equation (ii)

m=

m=

putting the value of a,b and m in equation (i) gives us

-12 = + c

-12=-12 + c

adding 12 to both sides

-12 +12 = -12 +12 +c

0 =c

or c=0

as the value of the constant is zero which gives us that it is a linear equation and this order pair satisfies the system of linear equations.