7.. SUPPLEMENTARY PROBLEMS FOR CHAPTER Supplementary Problems for Chapter 7 EXERCISE 7. A certain student always arrives the day before the te

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1 Modeling the Dynamics of Life: Calculus and for Life Scientists Frederick R. Adler cfrederick R. Adler, 22 Department of Mathematics and Department of Biology, University of Utah, Salt Lake City, Utah 842

2 7.. SUPPLEMENTARY PROBLEMS FOR CHAPTER Supplementary Problems for Chapter 7 EXERCISE 7. A certain student always arrives the day before the test clutching the practice problems, and asks about each with probability.6. a. Under what conditions is the binomial distribution appropriate? b. Under these conditions, what is the probability that the rst problem the student asks about is problem 4? c. What is the probability that this student asks about exactly 4 out of the rst 6? d. What are the expectation and variance of the number asked about if there are 8 practice problems? EXERCISE 7.2 A second student frequently appears after the rst has left. This student also asks about 6% of the problems, but asks about a particular problem with probability.8 if the rst student does not. a. Is the behavior of these students independent? How could you tell? b. Give the entire joint distribution. What is the probability that neither asks about problem 5? c. Draw the marginal distribution for the second student. d. Draw the conditional distributions for the second student. e. Find the covariance if asking is worth - and not asking is worth 2. EXERCISE 7.3 During a drizzle, raindrops hit a given square meter at a rate of 2.5 per minute. a. What is the expected number of raindrops after 3 seconds? b. What is the probability that exactly 5 raindrops hit in 3 seconds? c. What is the probability that no raindrops hit in seconds? d. If the drizzle lasts one hour, would you be surprised to nd a square meter with exactly 7 wet spots? EXERCISE 7.4 During a particularly wet spring, a rainy day follows a rainy day with probability.8, and a rainy day follows a dry day with probability.5. a. Write a discrete-time dynamical system describing this model. b. Draw the distributions that describe the weather on one day conditional on the weather of the previous day. c. Find the fraction of rainy days after a long time. d. Draw the joint distribution of weather on consecutive days after a long time. EXERCISE 7.5 A band is composed of 4 guitarists and a drummer. The drummer only knows how to keep a steady beat and each guitarist knows only the A and D chords. Each time the drummer hits the beat, each guitarist plays an A chord with probability.75 and a D chord with probability.25. a. Under what conditions is the binomial distribution appropriate for describing the number of A chords played on a given beat? b. What is the expectation and variance of the number of guitarists playing that A chord on a given beat? c. If the binomial distribution is appropriate, nd the probability that exactly 2 guitarists hit the A chord. d. Find the probability that all four guitarists play the same chord. e. Find the probability that all four guitarists play the same chord on exactly 2 out of 3 consecutive beats. EXERCISE 7.6 People attending a concert by this band are observed to be leaving at a rate of 3 per minute.

3 26 a. Under what conditions is the Poisson distribution appropriate for describing the number of people who left during a given minute? b. Under these conditions, nd the probability that nobody leaves during a period of length t and sketch the function. c. Find the expected number and the variance of people leaving during a given minute. d. Find the probability that exactly 3 people leave during a given minute. e. Find the probability that exactly 3 people leave during a two minute interval. EXERCISE 7.7 The remaining throngs show wild enthusiasm after a song with probability.3, mild enthusiasm with probability.2, and hurl insults with probability.5. The band reduces the volume by 2% in response to wild enthusiasm, by % in response to mild enthusiasm, and increases volume by 2% in response to insults. Suppose the volume begins at. a. Find the expected volume after one song. b. Would you expect the band to be louder or quieter at the end of a long concert? EXERCISE 7.8 After a particularly demoralizing performance, two guitarists decide to start working together. Recall that they know only two chords, A and D. a. If they began by playing chords independently, give the joint distribution. b. If the second player plays an A with probability 9% when the rst player does, give the joint distribution. c. If playing an A is worth point and a D is worth, nd the correlation. d. Draw the marginal and conditional distributions in this case. EXERCISE 7.9 An aspiring worm biologist decides to use a random procedure to stock his lab. He chooses among three catalogues, Baxter, Fisher and Sigma (). He orders from Baxter with probability.2, from Fisher with probability.7, and from Sigma with probability.. Products are delivered with probability.9 from Baxter,.95 from Fisher, and.8 from Sigma. a. Find the probability a product is delivered. b. Find the probability that a product that was not delivered was ordered from Fisher. c. Find the joint distribution. d. What strategy would maximize the probability of delivery? EXERCISE 7. A population switches between phases of growth (G) and decline (D) in subsequent years according to Pr(GjD) = :5 and Pr(DjG) = :5. The per capita growth rate is.2 during growth and.7 during decline. a. If the population grows in the rst year, nd the expected size after 2 years. b. If the population grows the rst year, what is the expected time until the rst decline? Sketch the distribution. c. Will this population grow or decline in the long run? d. Find the joint distribution and covariance between years after a long time. EXERCISE 7. A staining technique has a 2% chance of success. Let Sn be the random variable representing the number of successful stainings in n attempts. a. What are the conditions for Sn to have a binomial distribution? What are the parameters? b. Find E(S ) and Pr(S = 2).

4 7.. SUPPLEMENTARY PROBLEMS FOR CHAPTER 7 27 EXERCISE 7.2 Under the same conditions as in the previous problem, let T be a random variable representing the number of the rst successful attempt. For example, T = 2 means that the rst try was a failure and the second a success. a. Graph the probability distribution of T. b. Find the expectation and variance of T. EXERCISE 7.3 The density of bacterial colonies on a circular dish of radius 3 centimeters is.2 colonies per square centimeter. Let B be a random variable describing the number of colonies in the whole dish. a. Find the approximate probability of there being a colony in square millimeter. b. Find the expectation and variance of B. c. Find the probability of there being no colonies on the whole dish. EXERCISE 7.4 The woodpecker Picoides vilosus enjoys eating both the wood-boring tipulid (Ctenophora vittata) and the hickory bark beetle (Scolytus quadrispinosus). During any given minute, the woodpecker eats one tipulid with probability.5 and none with probability.5. Independently, during each minute, it eats one beetle with probability.25 and none with probability.75. a. Sketch the distribution describing the time when the rst beetle is eaten. b. Find the expected time, B, when the rst beetle is eaten. c. Find the probability that no tipulids had been eaten at or before B, and the expected number of tipulids eaten by time B. EXERCISE 7.5 A measurement X has normal distribution with mean and variance 9 and an independent measurement Y has normal distribution with mean 2 and variance 4. a. Find Pr(X ). b. Find E(X + Y ) and Var(X + Y ). c. Find Pr(X + Y 3). EXERCISE 7.6 Bubbles are produced by a chemical reaction at a rate of.8 per minute. Two assiduous students observe 2 bubbles during the rst minute, during the second minute, and fail to pay attention during the third minute because they are studying math. They then knuckle down and pay attention for the next thirty minutes. a. What is the probability that they didn't miss anything during that third minute? What is the probability that they missed 2 or more bubbles? b. What are the expectation and variance of the number of bubbles seen during the 3 minutes of paying attention? c. Write the formula for the probability of exactly 5 bubbles in 3 minutes. d. Two other students pay attention for only 5 minutes and multiply their results by 2. What is the expectation and variance for them? Explain the dierence. EXERCISE 7.7 Because students have proven unable to pay close attention to chemical reactions, the professor designs a new version of the experiment where bubbles do not pop. Students check for bubbles once per minute, record whether there are any (but don't count them), and pop all the ones that are there. The probability of at least one bubble during a minute is.83. a. What is the probability that the rst bubble is seen on the fth minute? b. Find the expected number of students in a class of 3 who would see no bubble during the rst 4 minutes. What is the variance? What does it mean?

5 28 c. What is the probability that a particular team of students will record bubbles in exactly 25 minutes out of 3? d. Estimate the probability of 25 or more minutes with bubbles during 3 minutes (no factorials allowed). Sketch the probability distribution and the indicate the probability you estimated. EXERCISE 7.8 Students begin measuring two numbers each minute, the number of bubbles B and the temperature T. B can take on only two values and, T can take on the values 2 and 3. The joint distribution is B = B = T = 2.. T = a. Find the marginal distributions. b. Find and sketch the conditional distribution of B when T = 2 and when T = 3. c. Find the covariance, and say what you would do to make this a more useful statistic. EXERCISE 7.9 Two mathematicians propose models for the growth of an elk population. The rst says that the number increases by B, where B is a random variable with a Poisson distribution and mean 5. The second says that the population increases by exactly 4% each year. Suppose the actual population begins at and is followed for years, and is found to be year t number of elk year t number of elk a. Sketch data that could result from each of the models. b. What is the expected number of elk and the variance after years according to the models? c. Which model predicts that the population will grow faster in the long run? d. What are the strengths and weaknesses of the two models in making sense of the data? e. If you had to construct a model of this population, what factors would you like to include?

6 7.. SUPPLEMENTARY PROBLEMS FOR CHAPTER a. If she asks about them independently. b. This is the geometric distribution, with value Pr(T = 4) = :4 3 :6 = :384. c. This is the binomial distribution, with value Pr(N = 4) = b(4; 6; :6) = 6 :6 4 :4 2 :3: 4 d. The expectation is np = 4:8 and the variance is np( p) = :92.! 7.2. a. Not independent because the conditional probability is not equal to the unconditional probability. b. To give the joint distribution for problem 5, let A represent the event that the rst student asks, N the event that she does not and so forth. A N A N Ask c. Conditional on first student asking Don t ask Conditional on first student not asking d. Ask Don t ask Ask Don t ask e. Let P be a random variable representing the number of points (- or 2) gotten by the rst student, similarly for the second. We have that E(P ) = E(P 2 ) = :6 ( ) + :4 2 = :2. Then Cov(P ; P 2 ) = E(P P 2 ) E(P )E(P 2 ) = :28 + :32 ( 2) + :32 ( 2) + :8 4 :2 :2 = :72: The negative value makes sense because the second student tends to ask when the rst does not.

7 a. Using the Poisson distribution, we nd E(N) = t = 2:5 :5 = 6:25. b. p(5; 6:25) = 6:25 5 e 6:25 =5! :53. c. In this case = 2:5 :67 = 2:8. The probability of drops is e 2:8 :25. d. The expected number in an hour is 2:5 6 = 75. The variance is then 75 and the standard deviation is p 75 27:4. The value 7 is within two standard deviations of 75, and wouldn't be too surprising a. Let p t = Pr(rainy on day t). Then p t+ = :8p t + :5( p t ): Conditional on rain the previous day Conditional on no rain the previous day Rain b. c. The equilibrium is.7. Dry Rain Dry d. R t D t R t D t a. They must play independently.! b. Expected number of A's is 3, the variance is.75. c. b(2; 4; :75) = 4 : :25 2 :2. d. They could either all play A or all play D, so the probability is b(; 4; :75) + b(4; 4; :75) = : :25 4 :32: e. This is b(2; 3; :32) = 3 : :68 :2.! 7.6. a. The people leave independently. b. is e 3t.

8 7.. SUPPLEMENTARY PROBLEMS FOR CHAPTER Time c. The expected number is E(N) = 3 and the variance is the same. d. p(3; 3) = 3 3 e 3 =3! :224. e. p(3; 6) = 6 3 e 6 =3! : a. The volume is reduced to 8 with probability.3, to 9 with probability.2, and increased to 2 with probability.5. The expectation is 8 :3 + 9 :2 + 2 :5 = 2. b. Because this is a multiplicative process, we must use the geometric mean. Let V be the random variable representing the change in volume (.8 with probability.3,.9 with probability.2, and.2 with probability.5). Then the geometric mean is e E(ln V ) = e ln(:8):3+ln(:9):2+ln(:2):5 = e :3 :3: Because this is greater than, the band will, on average, become louder by about 3% per song a. Supposing that they still play A with probability.75 and D with probability.25, the joint distribution is A D A D b. A D A D c. The expected value for each is.75. The expectation of the product is.675, so the covariance is.25. The variance for each is.875, so the correlation is.6. Marginal distribution Conditional on first playing A d. A D A D

9 82 Conditional on first playing D A D 7.9. a. Let D be the event \delivered," B be the event \ordered from Baxter," F be the event \ordered from Fisher," be the event \ordered from Sigma". By the law of total probability, b. By Bayes' theorem, Pr(D) = Pr(DjB)Pr(B) + Pr(DjF)Pr(F) + Pr(Dj)Pr() = :9 :2 + :95 :7 + :8 : = :925: Pr(FjD) = Pr(DjF)Pr(F) Pr(D) = :95 :7 :939 :79: c. Let N be the event \not delivered." d. Ordering everything from Fisher. B F D N a. Pr(G t+ jg t ) = :95, so it goes from.2 to.44 with probability.95 and from.2 to.84 with probability.5. The expectation is.4. b. This is a geometric distribution, with q = :5. The expected time is Time c. In the long run, the probability of growth is.75 (found from the Markov chain). The geometric mean is.49. The population will grow by about 5% per year.

10 7.. SUPPLEMENTARY PROBLEMS FOR CHAPTER 7 83 d. G t D t G t D t The expectation of each is :2 :75 + :7 :25 = :75. The expectation of the product is :44 :725 + :84 :75 + :49 :225 = :93, and the covariance is a. The attempts must be independent. p = :2 and n = n. b. E(S ) = np = 2. Pr(S = 2) = b(2; ; :2) = 45 :2 2 :8 8 : a. Time b. T has a geometric distribution with q = :2. The expectation is 5 and the variance is a. About.2. b. The area is cm 2, so B has a Poisson distribution with = :2 28:27 3:39. Both expectation and variance are c. e 3:39 : a. This is a geometric distribution with q = : Time b. Expected time is 4 minutes. c. no tipulids eaten in 4 minutes is :5 4 = :625. The expected number eaten is

11 84 a. X Pr(X ) = Pr 3 3 = Pr(Z :33) = (:33) = :63: = Pr Z 3 b. E(X + Y ) = E(X) + E(Y ) = 3 and Var(X + Y ) = Var(X) + Var(Y ) = 3. c. The standard deviation is p 3 = 3:66. X + Y Pr(X + Y 3) = Pr 3:65 3: = Pr Z 3:65 = Pr(Z 4:74) = ( 4:74): This is not on the chart, but the computer says : a. This follows a Poisson distribution. The probability of is e :8 :65. The probability of 2 or more is.537. b. Mean and variance are both 54. c. Pr(5 bubbles) = 545 e 54 : 5! d. The mean is 54, but the variance is 8. Fewer observations lead to lower certainty a. Only.69. b. Expect only.25 students, or one every 5 times the experiment is run. The variance is also.25, meaning that this is pretty uncertain. c. Using the binomial distribution, this is.92. d. The normal approximation to the binomial has mean 24.9 and variance We need to compute the probability that this exceeds 24.5, or that Z > :945, which is Number 7.8.

12 7.. SUPPLEMENTARY PROBLEMS FOR CHAPTER 7 85 a. Pr(B = ) = :7, Pr(B = ) = :83 Pr(T = 2) = :2, Pr(T = 3) = :8. Marginal distribution with T= b. Number of bubbles Marginal distribution with T= Number of bubbles c. The covariance is.66. I'd probably convert to the correlation to make it more useful Population 3 2 Poisson immigration Constant per capita production a b. The Poisson model would have 5 elk, with variance 5. The second model would have 48 elk with no variance. c. In the long run, the second model would be faster because it grows exponentially rather than linearly. d. The Poisson model deals better with randomness, but the actual data seem to increase faster as time goes along, more consistent with the second model. e. I would try to include both stochastic immigration and stochastic reproduction in each year. Year