Answer:
a. k = 3
b. Cumulative distribution function X,
c. Probability when headway exceeds 2 seconds = 0.125
Probability when headway is between 2 and 3 seconds = 0.088
d. Mean value of headway = 1.5
Standard deviation of headway = 0.866
e. Probability that headway is within 1 standard deviation of the mean value = 0.9245
Step-by-step explanation:
From the information provided,
Let X be the time headway between two randomly selected consecutive cars (sec).
The known distribution of time headway is,
a. Value of k.
Since the distribution of X is a valid density function, the total area for density function is unity. That is,
So, the equation becomes,
b. For this problem, the cumulative distribution function is defined as :
Now,
Therefore the cumulative distribution function X is,
c. Probability when the headway exceeds 2 secs.
Using cdf in part b, the required probability is,
Probability when headway is between 2 seconds and 3 seconds
Using the cdf in part b, the required probability is,
≅ 0.088
d. Mean value of headway,
And,
The standard deviation of headway is,
≅ 0.866
e. Probability that headway is within 1 standard deviation of the mean value
From part b, F(x) = 0, if x ≤ 1