5 “Time headway” in traffic flow is the elapsed time between the time that one car fin-

Example 4.5 “Time headway” in traffic flow is the elapsed time between the time that one car fin-

  ishes passing a fixed point and the instant that the next car begins to pass that point.

  Let X5 the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow. The following pdf of X is essentially the one suggested in “The Statistical Properties of Freeway Traffic” (Transp. Res., vol. 11: 221–228):

  f (x) 5 e

  .15e 2.15(x2.5) x .5

  0 otherwise

  The graph of f (x) is given in Figure 4.4; there is no density associated with headway times less than .5, and headway density decreases rapidly (exponentially fast) as x increases from .5. Clearly, f(x) 0

  ; to show that

  a 冕 2`

  the calculus result 冕 a e dx 5 (1k)e

  2k

  f (x) .15e 2.15(x2.5) dx 5 .15e .075 e 3 2.15x dx 5 3 dx

  5 .15e .075 1 e 2(.15)(.5) 51

  Figure 4.4 The density curve for time headway in Example 4.5

  CHAPTER 4 Continuous Random Variables and Probability Distributions

  The probability that headway time is at most 5 sec is

  f (x) dx 5 3 .15e P(X 5) 5 3 2.15(x2.5) dx

  5 .15e .075 e 2.15x dx 5 .15e .075 e 3 2.15x a2

  5e .075 (2e 2.75 1e 2.075 ) 5 1.078(2.472 1 .928) 5 .491

  5 P(less than 5 sec) 5 P(X , 5)

  ■

  Unlike discrete distributions such as the binomial, hypergeometric, and nega- tive binomial, the distribution of any given continuous rv cannot usually be derived using simple probabilistic arguments. Instead, one must make a judicious choice of pdf based on prior knowledge and available data. Fortunately, there are some general families of pdf’s that have been found to be sensible candidates in a wide variety of experimental situations; several of these are discussed later in the chapter.

  Just as in the discrete case, it is often helpful to think of the population of inter- est as consisting of X values rather than individuals or objects. The pdf is then a model for the distribution of values in this numerical population, and from this model various population characteristics (such as the mean) can be calculated.

  EXERCISES Section 4.1 (1–10)

  1. The current in a certain circuit as measured by an ammeter is

  4. Let X denote the vibratory stress (psi) on a wind turbine blade

  a continuous random variable X with the following density

  at a particular wind speed in a wind tunnel. The article

  function:

  “Blade Fatigue Life Assessment with Application to VAWTS” (J. of Solar Energy Engr., 1982: 107–111) proposes

  .075x 1 .2 3 x 5

  f (x) 5 e the Rayleigh distribution, with pdf

  0 otherwise

  e 2x 2 (2u a. Graph the pdf and verify that the total area under the den- 2 )

  sity curve is indeed 1.

  b. Calculate P(X 4) . How does this probability compare

  0 otherwise

  to ? P(X , 4)

  as a model for the X distribution.

  c. Calculate and P(3.5 X 4.5) also P(4.5 , X) .

  a. Verify that f(x; u) is a legitimate pdf.

  2. Suppose the reaction temperature X (in 8C ) in a certain

  b. Suppose

  (a value suggested by a graph in the

  chemical process has a uniform distribution with

  A 5 25

  article). What is the probability that X is at most 200? Less

  and . B55

  than 200? At least 200?

  a. Compute . P(X , 0)

  c. What is the probability that X is between 100 and 200

  b. Compute . P(22.5 , X , 2.5)

  (again assuming u 5 100 )?

  c. Compute . P(22 X 3)

  d. Give an expression for P(X x) .

  d. For k satisfying , 25 , k , k 1 4 , 5 compute 5. A college professor never finishes his lecture before the end of P(k , X , k 1 4) .

  the hour and always finishes his lectures within 2 min after the

  3. The error involved in making a certain measurement is a con-

  hour. Let X5 the time that elapses between the end of the

  tinuous rv X with pdf

  hour and the end of the lecture and suppose the pdf of X is

  kx 2 e 0x2

  .09375(4 2 x 2 ) 22 x 2

  a. Sketch the graph of f(x).

  a. Find the value of k and draw the corresponding density

  b. Compute . P(X . 0)

  curve. [Hint: Total area under the graph of f(x) is 1.]

  c. Compute . P(21 , X , 1)

  b. What is the probability that the lecture ends within 1 min

  d. Compute . P(X , 2.5 or X . .5)

  of the end of the hour?

  4.2 Cumulative Distribution Functions and Expected Values

  c. What is the probability that the lecture continues beyond

  the hour for between 60 and 90 sec?

  d. What is the probability that the lecture continues for at

  f ( y) 5

  e 2 2 1 y 5 y 10

  least 90 sec beyond the end of the hour?

  6. The actual tracking weight of a stereo cartridge that is set to

  0 y , 0 or y . 10

  track at 3 g on a particular changer can be regarded as a con- tinuous rv X with pdf

  a. Sketch a graph of the pdf of Y.

  f (x) 5 e

  k[1 2 (x 2 3) 2 ]2x4

  b. Verify that 3 f ( y) dy 5 1 .

  0 2` otherwise

  c. What is the probability that total waiting time is at most

  3 min?

  a. Sketch the graph of f(x).

  d. What is the probability that total waiting time is at most

  b. Find the value of k.

  8 min?

  c. What is the probability that the actual tracking weight is

  e. What is the probability that total waiting time is between

  greater than the prescribed weight?

  3 and 8 min?

  d. What is the probability that the actual weight is within

  f. What is the probability that total waiting time is either

  .25 g of the prescribed weight?

  less than 2 min or more than 6 min?

  e. What is the probability that the actual weight differs from the prescribed weight by more than .5 g?

  9. Consider again the pdf of X5 time headway given in Example 4.5. What is the probability that time headway is

  7. The time X (min) for a lab assistant to prepare the equipment

  a. At most 6 sec?

  for a certain experiment is believed to have a uniform distri-

  b. More than 6 sec? At least 6 sec?

  bution with

  A 5 25 and

  B 5 35 .

  c. Between 5 and 6 sec?

  a. Determine the pdf of X and sketch the corresponding density curve.

  10. A family of pdf’s that has been used to approximate the dis-

  b. What is the probability that preparation time exceeds

  tribution of income, city population size, and size of firms is

  33 min?

  the Pareto family. The family has two parameters, k and , u

  c. What is the probability that preparation time is within

  both .0 , and the pdf is

  2 min of the mean time? [Hint: Identify m from the graph

  k u k

  of f(x).]

  f (x; k, u) 5 x k11

  u

  xu

  d. For any a such that

  25 , a , a 1 2 , 35 , what is the

  0 x,u

  probability that preparation time is between a and a12 min?

  a. Sketch the graph of

  8. In commuting to work, a professor must first get on a bus

  b. Verify that the total area under the graph equals 1.

  near her house and then transfer to a second bus. If the wait-

  c. If the rv X has pdf f (x; k, u) , for any fixed b.u , obtain

  ing time (in minutes) at each stop has a uniform distribution

  an expression for P(X b) .

  with A50 and B55 , then it can be shown that the total

  d. For u,a,b , obtain an expression for the probability

  waiting time Y has the pdf

  P(a X b) .