11 Two species are competing in a region for control of a limited amount of a certain

Example 4.11 Two species are competing in a region for control of a limited amount of a certain

  resource. Let X5 the proportion of the resource controlled by species 1 and

  suppose X has pdf

  which is a uniform distribution on [0, 1]. (In her book Ecological Diversity, E. C. Pielou calls this the “broken-stick” model for resource allocation, since it is analo- gous to breaking a stick at a randomly chosen point.) Then the species that controls the majority of this resource controls the amount

  1 12X if 0 X ,

  h(X) 5 max (X, 1 2 X) 5 μ

  The expected amount controlled by the species having majority control is then

  E[h(X)] 5 3 max(x, 1 2 x)

  f (x) dx 5 3 max(x, 1 2 x)

  1 dx

  For h(X), a linear function, E[h(X)] 5 E(aX 1 b) 5 aE(X) 1 b . In the discrete case, the variance of X was defined as the expected squared devia-

  tion from m and was calculated by summation. Here again integration replaces summation.

  CHAPTER 4 Continuous Random Variables and Probability Distributions

  DEFINITION

  The variance of a continuous random variable X with pdf f(x) and mean value m is

  s 2 2 X 2 5 V(X) 5 3 (x 2 m) f (x)dx 5 E[(X 2 m) ]

  The standard deviation (SD) of X is . s X 5 2V(X)

  The variance and standard deviation give quantitative measures of how much spread there is in the distribution or population of x values. Again s is roughly the size of

  a typical deviation from . Computation of m s 2 is facilitated by using the same short-

  cut formula employed in the discrete case.

  PROPOSITION

  V(X) 5 E(X 2 ) 2 [E(X)] 2

  Example 4.12 3 For X5 weekly gravel sales, we computed E(X) 5

  E(X 2 x 2 x 2 3 (1 2 x )53 2 f (x) dx 5 3 ) dx

  V(X) 5

  When h(X ) 5 aX 1 b , the expected value and variance of h(X ) satisfy the same

  properties as in the discrete case: E[h(X )] 5 am 1 b and V[h(X )] 5 a 2 s 2 .

  EXERCISES Section 4.2 (11–27)

  11. Let X denote the amount of time a book on two-hour reserve

  12. The cdf for X ( 5 measurement error) of Exercise 3 is

  is actually checked out, and suppose the cdf is

  F(x) 5 d 1 2 32 a4x 2

  1 3 x 3

  b 22 x , 2

  F(x) 5 d 0x,2

  Use the cdf to obtain the following:

  a. Compute . P(X , 0)

  a. P(X 1)

  b. Compute . P(21 , X , 1)

  b. P(.5 X 1)

  c. Compute . P(.5 , X)

  c. P(X . 1.5)

  d. Verify that f(x) is as given in Exercise 3 by obtaining

  d. The median checkout duration | m [solve .5 5 F(m |)]

  F r(x) .

  e. Verify that m |50 .

  e. F r(x) to obtain the density function f(x)

  f. E(X)

  13. Example 4.5 introduced the concept of time headway in g. V(X) and s X traffic flow and proposed a particular distribution for X5

  h. If the borrower is charged an amount h(X ) 5 X 2 when

  the headway between two randomly selected consecutive

  checkout duration is X, compute the expected charge

  cars (sec). Suppose that in a different traffic environment,

  E[h(X)].

  the distribution of time headway has the form

  4.2 Cumulative Distribution Functions and Expected Values

  k

  19. Let X be a continuous rv with cdf

  F(x) 5 μ c1 1 lna

  x

  4 x bd 0 , x 4

  a. Determine the value of k for which f(x) is a legitimate pdf. b. Obtain the cumulative distribution function.

  1 x.4

  c. Use the cdf from (b) to determine the probability that

  [This type of cdf is suggested in the article “Variability in

  headway exceeds 2 sec and also the probability that

  Measured Bedload-Transport Rates” (Water Resources

  headway is between 2 and 3 sec.

  Bull., 1985: 39–48) as a model for a certain hydrologic vari-

  d. Obtain the mean value of headway and the standard

  able.] What is

  deviation of headway.

  a. P(X 1) ?

  e. What is the probability that headway is within 1 standard

  b. P(1 X 3) ?

  deviation of the mean value?

  c. The pdf of X?

  14. The article “Modeling Sediment and Water Column

  20. Consider the pdf for total waiting time Y for two buses

  Interactions for Hydrophobic Pollutants” (Water Research,

  1984: 1169–1174) suggests the uniform distribution on the

  interval (7.5, 20) as a model for depth (cm) of the bioturba-

  tion layer in sediment in a certain region.

  f ( y) 5 e 2 1

  2 y 5 y 10

  a. What are the mean and variance of depth?

  b. What is the cdf of depth?

  0 otherwise

  c. What is the probability that observed depth is at most

  introduced in Exercise 8.

  10? Between 10 and 15?

  a. Compute and sketch the cdf of Y. [Hint: Consider sepa-

  d. What is the probability that the observed depth is within

  rately and in 0y,5 5 y 10 computing F(y). A

  1 standard deviation of the mean value? Within 2 stan-

  graph of the pdf should be helpful.]

  dard deviations?

  b. Obtain an expression for the (100p)th percentile. [Hint:

  15. Let X denote the amount of space occupied by an article

  Consider separately 0 , p , .5 and .5 , p , 1 .]

  placed in a 1- ft 3 packing container. The pdf of X is

  c. Compute E(Y ) and V(Y ). How do these compare with the

  8 90x expected waiting time and variance for a single bus when (1 2 x) 0 , x , 1

  f (x) 5 e the time is uniformly distributed on [0, 5]?

  0 otherwise

  21. An ecologist wishes to mark off a circular sampling region

  a. Graph the pdf. Then obtain the cdf of X and graph it.

  having radius 10 m. However, the radius of the resulting

  b. What is P(X .5) [i.e., F(.5)]?

  region is actually a random variable R with pdf

  c. Using the cdf from (a), what is P(.25 , X .5) ? What

  is ? P(.25 X .5)

  f (r) 5 u 4

  [1 2 (10 2 r) 2 ] 9 r 11

  d. What is the 75th percentile of the distribution?

  e. Compute E(X ) and s X .

  0 otherwise

  f. What is the probability that X is more than 1 standard

  What is the expected area of the resulting circular region?

  deviation from its mean value?

  22. The weekly demand for propane gas (in 1000s of gallons)

  16. Answer parts (a)–(f ) of Exercise 15 with X5 lecture time

  from a particular facility is an rv X with pdf

  past the hour given in Exercise 5.

  17. Let X have a uniform distribution on the interval [A, B].

  f (x) 5 2 a1 2 2 b1x2 u x

  a. Obtain an expression for the (100p)th percentile.

  b. Compute E(X ), V(X ), and s X .

  0 otherwise

  c. For n, a positive integer, compute E(X n ) .

  a. Compute the cdf of X.

  18. Let X denote the voltage at the output of a microphone, and

  b. Obtain an expression for the (100p)th percentile. What is

  suppose that X has a uniform distribution on the interval

  the value of ? | m

  from 21 to 1. The voltage is processed by a “hard limiter”

  c. Compute E(X) and V(X ).

  with cutoff values 2.5 and .5, so the limiter output is a ran-

  d. If 1.5 thousand gallons are in stock at the beginning of

  dom variable Y related to X by if Y5X |X| .5, Y 5 .5 if

  the week and no new supply is due in during the week,

  X . .5 , and Y 5 2.5 if

  X , 2.5 .

  how much of the 1.5 thousand gallons is expected to be

  a. What is P(Y 5 .5) ?

  left at the end of the week? [Hint: Let h(x) 5 amount

  b. Obtain the cumulative distribution function of Y and

  left when demand 5x .]

  graph it.

  CHAPTER 4 Continuous Random Variables and Probability Distributions

  23. If the temperature at which a certain compound melts is a

  Although X is a discrete random variable, suppose its distri-

  random variable with mean value 1208C and standard devi-

  bution is quite well approximated by a continuous distribu-

  ation 28 C , what are the mean temperature and standard

  tion with pdf f(x) 5 k(1 1 x2.5) 27 for x0 .

  deviation measured in ? [Hint: .] 8F 8F 5 1.88 C 1 32

  a. What is the value of k?

  24. Let X have the Pareto pdf

  b. Graph the pdf of X. c. What are the expected value and standard deviation of

  k u k total medical expenses?

  f (x; k, u) 5

  u xu

  x k11

  d. This individual is covered by an insurance plan that

  0 x,u

  entails a 500 deductible provision (so the first 500 worth of expenses are paid by the individual). Then the

  introduced in Exercise 10.

  plan will pay 80 of any additional expenses exceed-

  a. If , k.1 compute E(X).

  ing 500, and the maximum payment by the individual

  b. What can you say about E(X) if k51 ?

  (including the deductible amount) is 2500. Let Y

  c. If , k.2 show that V(X) 5 ku 2 (k 2 1) 22 (k 2 2) 21 .

  denote the amount of this individual’s medical

  d. If k52 , what can you say about V(X)?

  expenses paid by the insurance company. What is the

  e. What conditions on k are necessary to ensure that E(X n )

  expected value of Y?

  is finite?

  [Hint: First figure out what value of X corresponds to

  25. Let X be the temperature in 8 C at which a certain chemical

  the maximum out-of-pocket expense of 2500. Then

  reaction takes place, and let Y be the temperature in 8 F (so

  write an expression for Y as a function of X (which

  Y 5 1.8X 1 32 ).

  involves several different pieces) and calculate the

  a. If the median of the X distribution is , show that m |

  expected value of this function.]

  1.8m | 1 32 is the median of the Y distribution.

  27. When a dart is thrown at a circular target, consider the loca-

  b. How is the 90th percentile of the Y distribution related to

  tion of the landing point relative to the bull’s eye. Let X be the

  the 90th percentile of the X distribution? Verify your

  angle in degrees measured from the horizontal, and assume

  conjecture.

  that X is uniformly distributed on [0, 360]. Define Y to be the

  c. More generally, if Y 5 aX 1 b , how is any particular

  transformed variable Y 5 h(X) 5 (2p360)X 2 p , so Y is

  percentile of the Y distribution related to the correspon-

  the angle measured in radians and Y is between 2p and . p

  ding percentile of the X distribution?

  Obtain E(Y) and s Y by first obtaining E(X) and s X , and then

  26. Let X be the total medical expenses (in 1000s of dollars)

  using the fact that h(X) is a linear function of X.

  incurred by a particular individual during a given year.