Solutions Modeling Dynamics of Life 3ed Adler - Chapter 7.7

7.7.1 The following figures show the results of simulations of the Poisson process with the given value of the rate λ. Find the number of hits per second in each of the simulated seconds. What is the average number of hits per second? How closely does this match what you would expect from the Poisson distribution? λ=1.5. ...
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7.7.2 The following figures show the results of simulations of the Poisson process with the given value of the rate λ. Find the number of hits per second in each of the simulated seconds. What is the average number of hits per second? How closely does this match what you would expect from the Poisson distribution? λ=2.0. ...
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7.7.3 The following figures show the results of simulations of the Poisson process with the given value of the rate λ. Find the number of hits per second in each of the simulated seconds. What is the average number of hits per second? How closely does this match what you would expect from the Poisson distribution? λ=0.5. ...
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7.7.4 The following figures show the results of simulations of the Poisson process with the given value of the rate λ. Find the number of hits per second in each of the simulated seconds. What is the average number of hits per second? How closely does this match what you would expect from the Poisson distribution? λ=0.8. ...
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7.7.5 Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution with the given value of λ and t =1 by plotting both distributions. The simulation in Exercise 1 with λ=1.5.
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7.7.6 Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution with the given value of λ and t =1 by plotting both distributions. The simulation in Exercise 2 with λ=2.0.
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7.7.7 Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution with the given value of λ and t =1 by plotting both distributions. The simulation in Exercise 3 with λ=0.5.
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7.7.8 Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution with the given value of λ and t =1 by plotting both distributions. The simulation in Exercise 4 with λ=0.8.
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7.7.10 Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution with the given value of λ and t =1 by plotting both distributions. The simulation in Exercise 4 with λ=0.8. Regroup the results into ten intervals with length 2 s, and compare with the Poisson distribution with λ=0.8 and t =2.
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7.7.10 Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution with the given value of λ and t =1 by plotting both distributions. The simulation in Exercise 4 with λ=0.8. Regroup the results into ten intervals with length 2 s, and compare with the Poisson distribution with λ=0.8 and t =2.
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7.7.11 The following figures show the results of simulations that do not follow the assumptions of the Poisson process. Can you identify how they differ? ...
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7.7.12 The following figures show the results of simulations that do not follow the assumptions of the Poisson process. Can you identify how they differ? ...
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7.7.13 The following figures show the results of simulations that do not follow the assumptions of the Poisson process. Can you identify how they differ? ...
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7.7.14 The following figures show the results of simulations that do not follow the assumptions of the Poisson process. Can you identify how they differ? ...
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7.7.15 Find Λ, the expectation, the variance, the standard deviation, the coefficient of variation, and the mode of Poisson distributions with the following values of λ and t. Molecules leave a cell at rate λ=0.3/s and are observed for t =3 s.
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7.7.16 Find Λ, the expectation, the variance, the standard deviation, the coefficient of variation, and the mode of Poisson distributions with the following values of λ and t. Phone calls arrive at a rate of λ=0.2/h and are monitored for t =9 h.
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7.7.17 Find Λ, the expectation, the variance, the standard deviation, the coefficient of variation, and the mode of Poisson distributions with the following values of λ and t. Cosmic rays hit an organism at a rate of 1.2/day and are monitored for one week.
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7.7.18 Find Λ, the expectation, the variance, the standard deviation, the coefficient of variation, and the mode of Poisson distributions with the following values of λ and t. Dandelion seeds fall into a garden with area 4.0 ... with an average density of 0.9/ ....
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7.7.19 Find the probabilities of the following events. Molecules leave a cell at rate λ=0.3/s. What is the probability that exactly two have left by the end of the third second?
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7.7.20 Find the probabilities of the following events. Phone calls arrive at a rate of λ=0.2/h. What is the probability that there are exactly five calls in 9 h?
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7.7.21 Find the probabilities of the following events. Cosmic rays hit an organism at a rate of 1.2/day. What is the probability of being hit ten times in a week?
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7.7.22 Find the probabilities of the following events. Dandelion seeds fall into a garden with an average density of 0.9/ .... What is the probability that three fall into a 4.0-... vegetable garden?
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7.7.23 Find and sketch the Poisson distribution associated with the given rate λ and duration t, and use it to compute the requested probability. Molecules leave a cell at rate λ=0.3/s. What is the probability that four or fewer have left by the end of the third second?
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7.7.24 Find and sketch the Poisson distribution associated with the given rate λ and duration t, and use it to compute the requested probability. Phone calls arrive at a rate of λ=0.2/h. What is the probability that there are five or more calls in 9 h?
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7.7.25 Find and sketch the Poisson distribution associated with the given rate λ and duration t, and use it to compute the requested probability. Cosmic rays hit an organism at a rate of 1.2/day. What is the probability of being hit between five and ten times(inclusive) in a week?
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7.7.26 Find and sketch the Poisson distribution associated with the given rate λ and duration t, and use it to compute the requested probability. Dandelion seeds fall into a garden with an average density of 0.9/m2. What is the probability that between two and five seeds (inclusive) fall into a 4.0 m2 vegetable garden?
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7.7.28 Molecules leave a cell at rate λ=0.3/s. Let N(t) be the random variable measuring the number of cells that have left as a function of t. Consider times t between 0 and 10. Compute and graph CV[N(t)] as a function of time.
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7.7.28 Molecules leave a cell at rate λ=0.3/s. Let N(t) be the random variable measuring the number of cells that have left as a function of t. Consider times t between 0 and 10. Compute and graph CV[N(t)] as a function of time.
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7.7.29 Molecules leave a cell at rate λ=0.3/s. Let N(t) be the random variable measuring the number of cells that have left as a function of t. Consider times t between 0 and 10. Compute and graph Pr[N(t)=1] as a function of time. Find the maximum. Why does this graph increase and then decrease?
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7.7.30 Molecules leave a cell at rate λ=0.3/s. Let N(t) be the random variable measuring the number of cells that have left as a function of t. Consider times t between 0 and 10. Compute and graph Pr[N(t)=2] as a function of time. Find the maximum. Why does the maximum occur later than the maximum of Pr[N(t)=1]?
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7.7.31 Find the parameter _ for the Poisson distribution that describes the given process. Cells will mutate when hit independently by cosmic rays (at rate 0.3/day) or by X-rays (at rate 0.2/day). Cells are hit by rays for 1 week. Use random variables ...and N to describe the number of mutations caused by cosmic rays, X-rays, and both types of rays together. What are their distributions? Check that ...and ...
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7.7.32 Find the parameter _ for the Poisson distribution that describes the given process. A professor is interrupted independently by phone calls (at rate 1.3/h), by students with questions (at rate 0.6/h), and by colleagues (at rate 0.3/h). How many interruptions might she expect during an 8-h day? What is the expected time between phone calls, between students, between colleagues, and between interruptions of all kinds?
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7.7.33 In each of the following cases, find the probability exactly with the binomial distribution, and compare your result with what you find with the Poisson approximation. Each cell in a culture of 16 cells has a probability of 0.1 of dying. Find and approximate the probability that exactly one cell dies.
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7.7.34 In each of the following cases, find the probability exactly with the binomial distribution, and compare your result with what you find with the Poisson approximation. Assume you get one call in a given hour with probability 0.2 and zero calls with probability 0.8. Find and approximate the probability of exactly two calls in 10 h.
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7.7.35 The probabilities for the Poisson distribution can be derived by solving differential equations. Let ...be the probability of exactly i events by time t, assuming an underlying rate of λ. Write ...in terms of ... (think of the two ways there could have been one molecule at time t + Δt).
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7.7.36 The probabilities for the Poisson distribution can be derived by solving differential equations. Let ...be the probability of exactly i events by time t, assuming an underlying rate of λ. Move stuff around so your formula looks like the derivative of ...and take the limit as Δt→0.
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7.7.38 The probabilities for the Poisson distribution can be derived by solving differential equations. Let ...be the probability of exactly i events by time t, assuming an underlying rate of λ. ...
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7.7.38 The probabilities for the Poisson distribution can be derived by solving differential equations. Let ...be the probability of exactly i events by time t, assuming an underlying rate of λ. ...
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7.7.40 We can also use differential equations to derive the formulas for the expectation and variance of the Poisson distribution. Let the variance in the number of events that have occurred in a Poisson process be V (t). Using the definition of a probabilistic rate to find a formula for V (t + Δt), write and solve a differential equation for V (t).
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7.7.40 We can also use differential equations to derive the formulas for the expectation and variance of the Poisson distribution. Let the variance in the number of events that have occurred in a Poisson process be V (t). Using the definition of a probabilistic rate to find a formula for V (t + Δt), write and solve a differential equation for V (t).
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7.7.41 Genes in different organisms have different rates of mutation. Compute the following values and probabilities. A gene has a mutation rate of 0.002 mutations per generation. Find the expected number of mutations, the variance, the probability of zero mutations, and the probability of exactly one mutation in a period of 2000 generations.
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7.7.42 Genes in different organisms have different rates of mutation. Compute the following values and probabilities. An important gene has a mutation rate of 0.0004 mutations per generation. Find the expected number of mutations, the variance, the probability of zero mutations, and the probability of exactly one mutation in a period of 2000 generations.
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7.7.43 Genes in different organisms have different rates of mutation. Compute the following values and probabilities. A gene has a mutation rate of 0.002 mutations per generation. How many generations would it take for the expected number of mutations to be greater than 1? How many generations would it take before the probability of zero mutations is less than 0.001?
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7.7.44 Genes in different organisms have different rates of mutation. Compute the following values and probabilities. An important gene has a mutation rate of 0.0004 mutations per generation. How many generations would it take for the expected number of mutations to be greater than 1? How many generations would it take before the probability of zero mutations is less than 0.0001?
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7.7.45 When two species of organisms have genetically diverged, the number of mutations distinguishing particular genes is the sum of the number in one organism and the number in the other. For the following cases, find the expected number of mutations distinguishing the species. Two populations of fruit flies diverged 10,000 years ago. In the first populations, a gene has a mutation rate of 0.002 mutations per generation. In the second population, a gene has a mutation rate of 0.0004 mutations per generation. Generations in each population are 1 yr.
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7.7.46 When two species of organisms have genetically diverged, the number of mutations distinguishing particular genes is the sum of the number in one organism and the number in the other. For the following cases, find the expected number of mutations distinguishing the species. Two populations of flies diverged 1 million years ago. In the first population, a gene has a mutation rate of 1.5×... mutations per generation. In the second population, a gene has a mutation rate of 3.0×...mutations per generation. Generations in each population are 0.5 yr.
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7.7.47 In most genes, mutations in some sites (“synonymous sites”) have no effect on the protein produced, while those in other sites (“nonsynonymous sites”) do affect the protein. In general, mutation rates are higher in synonymous sites because changes are not removed by natural selection. A gene has 200 nonsynonymous sites and 100 synonymous sites. The synonymous sites have mutation rate 6.0×.../yr, while nonsynonymous sites have mutation rate 3.0×.../yr. What is the expected number of mutations of each type after 1 million years? What is the probability of no mutations of either type?
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7.7.48 In most genes, mutations in some sites (“synonymous sites”) have no effect on the protein produced, while those in other sites (“nonsynonymous sites”) do affect the protein. In general, mutation rates are higher in synonymous sites because changes are not removed by natural selection. A gene has 200 nonsynonymous sites and 100 synonymous sites. The synonymous sites have mutation rate 6.0×.../yr, while nonsynonymous sites have mutation rate 3.0×.../yr. What is the expected number of mutations of each type after 1 million years? What is the probability of no mutations of either type?
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7.7.49 Different environments can lead to different mutation rates. Use the sum rule for the Poisson distribution to find the total expected number of mutations during an experiment by finding the expected number over the entire experiment. Mutations accumulate at a rate of 1.3 per million nucleotides during the first year of a study, and at a rate of 2.2 per million nucleotides during the second year. The DNA is 4.7 million nucleotides long.
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7.7.50 Different environments can lead to different mutation rates. Use the sum rule for the Poisson distribution to find the total expected number of mutations during an experiment by finding the expected number over the entire experiment. Mutations accumulate at a rate of 0.3 per million nucleotides during the first 0.5 yr of a study, and at a rate of 3.2 per million nucleotides during the second 1.5 yr. The DNA is 4.7 million nucleotides long.
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7.7.51 The previous problem shows that the number of mutations follows a Poisson distribution even when the mutation rate changes. In fact, if the rate changes continuously, following some function λ(t), the number of mutations between time 0 and time t follows a Poisson distribution with mean ... Use this formula to analyze the following circumstances. Due to an increase in radiation levels, the mutation rate increases linearly from ...at ...Find the expected number of mutations over the course of 2 million years. What is the average mutation rate over this time?
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7.7.52 The previous problem shows that the number of mutations follows a Poisson distribution even when the mutation rate changes. In fact, if the rate changes continuously, following some function λ(t), the number of mutations between time 0 and time t follows a Poisson distribution with mean ... Use this formula to analyze the following circumstances. Due to a decrease in radiation, the mutation rate decreases linearly from ...at t = .... Find the expected number of mutations over the course of 2 million years. What is the average mutation rate over this time?
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7.7.52 The previous problem shows that the number of mutations follows a Poisson distribution even when the mutation rate changes. In fact, if the rate changes continuously, following some function λ(t), the number of mutations between time 0 and time t follows a Poisson distribution with mean ... Use this formula to analyze the following circumstances. Due to a decrease in radiation, the mutation rate decreases linearly from ...at t = .... Find the expected number of mutations over the course of 2 million years. What is the average mutation rate over this time?
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7.7.54 Suppose many gnats are flying around in a room. Each leaves independently with a probability that depends on the insect repellent tested. In each case:
a. Describe the random variable with a binomial distribution giving the probability that exactly k have left.
b. Find the Poisson distribution approximating the probability that exactly k have left and the expected number to leave.
c. Find the variance of the binomial random variable and the Poisson approximation. There are 50,000 gnats, and each leaves with probability 0.037.
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7.7.55 Suppose an organism would live forever if it weren’t for predators that attack at rate λ per year. Fortunately, only a fraction q of attacks are successful. Use the following steps to compute how long the organism will live. What is the probability that the organism has not been attacked at time t?
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7.7.56 Suppose an organism would live forever if it weren’t for predators that attack at rate λ per year. Fortunately, only a fraction q of attacks are successful. Use the following steps to compute how long the organism will live. What is the probability that the organism has been attacked once but survived?
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7.7.57 Suppose an organism would live forever if it weren’t for predators that attack at rate λ per year. Fortunately, only a fraction q of attacks are successful. Use the following steps to compute how long the organism will live. What is the probability that the organism has been attacked twice but survived?
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7.7.58 Suppose an organism would live forever if it weren’t for predators that attack at rate λ per year. Fortunately, only a fraction q of attacks are successful. Use the following steps to compute how long the organism will live. Write down a sum giving the probability that the organism is alive at time t.
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7.7.59 Suppose an organism would live forever if it weren’t for predators that attack at rate λ per year. Fortunately, only a fraction q of attacks are successful. Use the following steps to compute how long the organism will live. Write down the first few probabilities in a Poisson distribution with parameter (1 − q. What do they add up to?
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7.7.60 Suppose an organism would live forever if it weren’t for predators that attack at rate λ per year. Fortunately, only a fraction q of attacks are successful. Use the following steps to compute how long the organism will live. Use this last fact to come up with the probability that the organism is alive at time t. What is the average lifetime? Could you have guessed it without doing any calculations?
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7.7.61 Suppose T is an exponentially distributed random variable with parameter λ=0.3 that describes the waiting time between events. If the nth event occurs at time t, the n + 1st event occurs at time t + T . The updating function describing this process is g(t)=t + T, giving the time of the next event as a stochastic function of the time of the previous event.
a. Starting from an initial condition of 0, simulate this updating function for 20 steps. How many events occurred by time 20?
b. Mark the two longest waits and their durations.
c. What is the expected waiting time between events? Sketch the line that should lie close to the graph of your solution.
d. The number of events at time 20 should be described by a Poisson distribution with parameter Λ=20λ. Plot the histogram of this distribution.
e. Using the number of events that occurred by time 20 in part a as k, compute the probability of exactly k events, fewer than k events, and more than k events. Indicate the areas associated with each of these events on your histogram.
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7.7.62 In the previous exercise, we simulated a Poisson process as a series of exponentially distributed waiting times. An alternative method uses the relation between the Poisson and binomial distributions. Suppose we wish to simulate a process with λ=1.5/min for 10 min.
a. Break each minute into n intervals for some large value of n, such as 100. What value of p will produce an average of 1.5 successes per minute?
b. Choose a series of 10n independent Bernoulli random variables that equal 1 with probability p. Each value of 1 corresponds to an event in the Poisson process.
c. Produce a graph showing when the events occur.
d. Why does this method fail to exactly reproduce the Poisson process?
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