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Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions

Exactly two remain. At what time is this probability a maximum? Starting with five molecules, each leaving with probability 0.2/min never to return, find and graph the above probabilities as functions of time.
Find the expected number remaining inside as a function of time. 40 molecules begin inside a cell. Each leaves independently with probability 0.2/min.
Two or fewer out of eight offspring are tall.Suppose that the allele A for height is dominant, meaning that plants with genotypes AA and Aa are tall, while those with genotype aa are short. If an Aa plant is crossed with another Aa plant, 3/4 of the offspring should be tall. Assuming that the other
Example 6.1.5 from Section 6.1, which illustrates stochastic immigration, was generated by adding two individuals with probability 0.5 and zero individuals with probability 0.5 for 100 generations. The results in the figure show final populations of 106 and 96.1. How can these results be described
Suppose p1 = 0 and p2 = 1. Find and graph the probability distribution for the total number inside. Find the expectation and the variance. Unbeknownst to the experimenter, a cell contains two different types of molecule, one which is inside with probability p1 and the other which is inside with
Suppose p1 = 0.6 and p2 = 0.8. Find and graph the probability distribution for the total number inside. Find the expectation and the variance (this can be written as the sum of two binomial random variables). Unbeknownst to the experimenter, a cell contains two different types of molecule, one
Suppose p1 = 0.25 and p2 = 0.75. Find and graph the probability distribution for the total number inside. Find the expectation and the variance (this can be written as the sum of two binomial random variables). Unbeknownst to the experimenter, a cell contains two different types of molecule, one
It turns out that all four molecules are different and that p1 = 0, p2 = 0.25, p3 = 0.75, and p4 = l. Find and graph the probability distribution for the total number inside. Find the expectation and the variance. Unbeknownst to the experimenter, a cell contains two different types of molecule, one
p = 0.5, n = 100. Compute the coefficient of variation of the binomial distribution in the above case.
p = 0.5, n = 25. Why is the coefficient of variation larger than in Exercise 5? Compute the coefficient of variation of the binomial distribution in the above case.
p = 0.9, n = 25. Compute the coefficient of variation of the binomial distribution in the above case.
p = 0.1, n = 25. Why is the coefficient of variation larger than Exercise 7? Compute the coefficient of variation of the binomial distribution in the above case.
Out of three offspring, one is tall, one is intermediate, and one is short. When there are more than two outcomes of a trial, the distribution of all possibilities is described by the multinomial distribution. Consider an additive pair of alleles A and a, where an offspring of a cross between two
The probability that the first success is on the third trial if each trial has a probability 0.2 of success. Compute the above probabilities. In each case, sketch the probability distribution and shade the associated area.
Events occur at rate 5.0/s. Find the probability that the first event occurs before t = 0.1 or after t = 0.5. Compute the above probabilities. In each case, sketch the probability distribution function and shade the area associated with the question.
The probability of success is q = 0.2. Find the mean, variance, standard deviation, coefficient of variation, and mode for a random variable T that follows a geometric distribution with the given probability of success.
The probability of success is q = 0.3. Find the mean, variance, standard deviation, coefficient of variation, and mode for a random variable T that follows a geometric distribution with the given probability of success.
The probability of success is q = 0.7. Find the mean, variance, standard deviation, coefficient of variation, and mode for a random variable T that follows a geometric distribution with the given probability of success.
The probability of success is q = 0.9. Find the mean, variance, standard deviation, coefficient of variation, and mode for a random variable T that follows a geometric distribution with the given probability of success.
The rate is λ = 0.2. Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ.
The rate is λ = 0.5. Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ.
The rate is λ = 1.5. Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ.
The rate is λ = 5.0. Find the mean, variance, standard deviation, coefficient of variation, median, and mode for a random variable T that follows an exponential distribution with the given rate λ.
We can verify that the cumulative distribution for the geometric distribution is Gt = 1 - (1 - q)t with a mathematical trick.1. Show thatfor any x by multiplying both sides by 1 - x and working out the algebra.2. Find(use x = l - q).
The probability that the first success is on the fifth trial if each trial has a probability 0.3 of success. Compute the above probabilities. In each case, sketch the probability distribution and shade the associated area.
Use this Taylor series to show thatfor any 0 We found that the function f(x) = 1/1 - x is equal to the Taylor seriesTf(x) = l + x + x2 + x3 + x4 + ...when 0
Differentiate the Taylor series term by term and use it to derive the expectation of a geometric random variable. We found that the function f(x) = 1/1 - x is equal to the Taylor series Tf(x) = l + x + x2 + x3 + x4 + ... when 0 < x < 1 (Example 3.7.10).
Find the probability that a molecule does not leave immediately. Show that the expected time to leave in this case is 1 + E(T). The expectation of the geometric distribution can be found using a clever trick.
The probability that the first success occurs on or before the fourth trial if each has a probability 0.2 of success. Compute the above probabilities. In each case, sketch the probability distribution and shade the associated area.
Why does R have the same probability distribution as T? The variance of the geometric distribution can be found with a clever trick, much like that in Exercises 27-30. Define the random variable R, which is 1 with probability q and 1 + T with probability 1 - q.
Solve for E(T2) and use the computational formula to find Var(T). The variance of the geometric distribution can be found with a clever trick, much like that in Exercises 27-30. Define the random variable R, which is 1 with probability q and 1 + T with probability 1 - q.
The expectation. Use integration by parts to compute the following statistics for an exponentially distributed random variable with parameter λ.
The variance. Use integration by parts to compute the following statistics for an exponentially distributed random variable with parameter λ.
A certain highly mutable gene has a 1.0% chance of mutating each time a cell divides. Suppose that there are 15 cell divisions between each pair of generations. What is the chance that the gene first mutates during the last division? What is the chance that the gene mutates at some point during the
A herd of lemmings is standing at the top of a cliff, and each jumps off with probability 0.2 each hour. What is the probability that a lemming first jumps on the fifth hour? What is the probability that a particular lemming has not jumped by the end of third hour? Find the probabilities of the
A molecule has a 5.0% chance of binding to an enzyme each second and remains permanently attached thereafter. Find the probability that it binds during the tenth second, and the probability that it binds on the tenth second or before. Find the probabilities of the above events.
The probability that the first success occurs on or before the third trial if each has a probability 0.3 of success. Compute the above probabilities. In each case, sketch the probability distribution and shade the associated area.
In tropical regions, growing caterpillars can be eaten with probability 0.15 each day. What is the probability that a caterpillar is eaten on the fourth day? If a caterpillar takes 25 days to develop, what is the probability it survives? Find the probabilities of the above events.
A light bulb blows out with probability 0.01 each day. What is the probability that it blows out on the 50th day? What is the probability it blows out after more than 200 days? Find the probabilities of the above events.
10% of some type of item are defective. Find the probability that the first defective item found is the fifth one inspected. What is the probability that a defective one is found if five are inspected? Find the probabilities of the above events.
Use the binomial distribution to find the expected number of purple-eyed flies (from a mutant parent) after n have been checked. How many must be checked for the expectation to equal 1? A lab is screening to find mutant flies. 10% of the offspring of mutant flies have purple eyes, while none of the
What is the expected number of mutant flies that must be inspected to find the first one with purple eyes? What is the expected number of purple-eyed flies that will be found if this many are checked? What is the probability that exactly one purple-eyed fly is found if this many are checked? A lab
Suppose that half of the parent flies are known to be mutants. Of 20 offspring flies inspected, none have purple eyes. What is the conditional probability that the parents are mutants? If parents are discarded if all 20 offspring have normal eyes, what fraction of mutant parents will be
Suppose that only 5% of the parent flies are known to be mutants. Of 20 flies inspected, none are found to be have purple eyes. What is the conditional probability that the parents are mutants? If parents are discarded if all 20 offspring have normal eyes, what fraction of mutant parents will be
Find the whole probability distribution. Inspecting for defective items until the first is found only works when all the items to be checked have the same probability of being defective. Suppose that items to be inspected had been deceptively sorted in the factory in order from best to worst. The
A molecule leaves a cell at rate λ = 0.3/s. What is the probability it has left by the end of the third second? By the end of the first millisecond? Compare your last answer with the definition of rate. Find the probabilities of the above events. Find the associated p.d.f., c.d.f., and
The probability that the first success occurs on or after the third trial if each has a probability 0.2 of success. Compute the above probabilities. In each case, sketch the probability distribution and shade the associated area.
A light bulb blows out at a rate of λ = 0.001/h. What is the probability that it blows out in less than 500 h? In less than 1 h? Compare your last answer with the definition of rate. Find the probabilities of the above events. Find the associated p.d.f., c.d.f., and survivorship function, and give
Phone calls arrive at a rate of λ = 0.2/h. What is the probability that there are no Calls in 10 h? What is the probability that a call arrives during the 45 s you spend in the bathroom? Find the probabilities of the above events. Find the associated p.d.f., c.d.f., and survivorship function, and
Raindrops hit a leaf at rate 7.3/min. What is the probability that the first one hits in less than 0.5 min? Find the probabilities of the above events. Find the associated p.d.f., c.d.f., and survivorship function, and give the expectation and variance.
A population of 100 bacteria are dying independently at rate λ = 2.0.1. Find the probability that a given bacterium is alive at time t.2. What distribution describes the population at time t? Find the expectation and variance of the number alive as functions of time.
Compute the following probabilities.1. The gene in Exercise 37 has not mutated after ten divisions. What is the probability that it mutates by the 15th division?2. The caterpillar in Exercise 40 is still alive after 10 days. What is the probability that it is eaten by day 25?3. The molecule in
If the length of time that cars remain on the freeway is an exponential distribution, what is the probability that the new a.s.d. exits first? Under what conditions might this occur? You are trapped behind an annoyingly slow driver (a.s.d.) in a long no-passing zone. A second a.s.d. merges in front
The probability that the first success occurs on or after the sixth trial if each trial has a probability 0.3 of success. Compute the above probabilities. In each case, sketch the probability distribution and shade the associated area.
Under what assumptions would you expect the new a.s.d. to exit first? Under what assumptions would you expect the new a.s.d. to exit second? You are trapped behind an annoyingly slow driver (a.s.d.) in a long no-passing zone. A second a.s.d. merges in front of the first, leaving you twice as
Solve the equation with separation of variables (Algorithm 5.2). Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years.
Graph the survivorship function. Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years.
Find the c.d.f. and p.d.f. Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years.
Find the probability it survives to age 2 conditional on surviving to age 1. Why might this value be different from that in Exercise 65? Suppose that some species of insect dies faster as it gets older according to rate of death = t, where t is the age in years.
Events occur at a rate of 0.5/s. Find the probability that the first event occurs between times 1.0 and 2.0. Compute the above probabilities. In each case, sketch the probability distribution function and shade the area associated with the question.
Events occur at a rate of 1.5/s. Find the probability that the first event occurs between times 0.2 and 1.0. Compute the above probabilities. In each case, sketch the probability distribution function and shade the area associated with the question.
Events occur at rate 0.2/s. Find the probability that the first event occurs before t = 1 or after t = 3. Compute the above probabilities. In each case, sketch the probability distribution function and shade the area associated with the question.
The following figures show the results of simulations of the Poisson process with the given value of the rate Î». 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 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. Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the
The following figures show the results of simulations that do not follow the assumptions of the Poisson process. Can you identify how they differ?1.2.3.4.
Molecules leave a cell at rate λ = 0.3/s and are observed for t = 3 s. Find Λ, the expectation, the variance, the standard deviation, the coefficient of variation, and the mode of Poisson distributions with the above values of λ and t.
Phone calls arrive at a rate of λ = 0.2/h and are monitored for r = 9 h. Find Λ, the expectation, the variance, the standard deviation, the coefficient of variation, and the mode of Poisson distributions with the above values of λ and t.
Cosmic rays hit an organism at a rate of 1.2/day and are monitored for one week. Find Λ, the expectation, the variance, the standard deviation, the coefficient of variation, and the mode of Poisson distributions with the above values of λ and t.
Dandelion seeds fall into a garden with area 4.0 m2 with an average density of 0.9/m2. Find Λ, the expectation, the variance, the standard deviation, the coefficient of variation, and the mode of Poisson distributions with the above values of λ and t.
Find the probabilities of the following events.1. 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?2. Phone calls arrive at a rate of λ = 0.2/h. What is the probability that there are exactly five calls in 9 h?3. Cosmic
The following figures show the results of simulations of the Poisson process with the given value of the rate Î». 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
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? 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? 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? 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? Find and sketch the Poisson distribution associated with the given rate λ. and duration t, and use it to compute the
Compute and graph E[N(t)] as a function of time. 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. 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? 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.
The following figures show the results of simulations of the Poisson process with the given value of the rate Î». 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
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]? 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
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 Mc, Mx, and N to describe the number of mutations caused by cosmic rays, X-rays, and both types of rays together. What are their
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
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. In the above case, 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. In the above case, find the probability exactly with the binomial distribution, and compare your result with what you find with the
Move stuff around so your formula looks like the derivative of P1(t) and take the limit as ∆t → 0. The probabilities for the Poisson distribution can be derived by solving differential equations. Let Pi (t) be the probability of exactly i events by time t, assuming an underlying rate of λ.
Recalling that P0(t) = e-λt, check that P1(t) = λte-λt is a solution of your equation with initial condition P1(0) = 0. The probabilities for the Poisson distribution can be derived by solving differential equations. Let Pi (t) be the probability of exactly i events by time t, assuming an
Use the same method to find an equation for P2(t) and check that
Let the expected number of events that have occurred in a Poisson process be E(t). Using the definition of a probabilistic rate to find a formula for E(t + ∆t), write and solve a differential equation for E(t). We can also use differential equations to derive the formulas for the expectation and
The following figures show the results of simulations of the Poisson process with the given value of the rate Î». 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
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 + ∆t), write and solve a differential equation for V(t). We can also use differential equations to derive the formulas for the
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. Genes in different organisms have different rates of mutation. Compute
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. Genes in different organisms have different rates of
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? Genes in different organisms have different
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? Genes in different organisms have
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. When two species of
Two populations of flies diverged 1 million years ago. In the first population, a gene has a mutation rate of 1.5 × 10-5 mutations per generation. In the second population, a gene has a mutation rate of 3.0 × 10-6 mutations per generation. Generations in each population are 0.5 yr. When two
A gene has 200 non synonymous sites and 100 synonymous sites. The synonymous sites have mutation rate 6.0 × 10-7/yr, while non synonymous sites have mutation rate 3.0 × 10-9/yr. What is the expected number of mutations of each type after 1 million years? What is the probability of no mutations of
A gene has 300 non synonymous sites and 150 synonymous sites. The synonymous sites have mutation rate 6.0 × 10-7/yr, while non synonymous sites have mutation rate 3.0 × 10-9/yr. What is the expected number of mutations of each type after 2 million years? What is the probability of no mutations of
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. Different environments can lead to different mutation rates. Use the sum rule for the Poisson
The simulation in Exercise 1 with λ = 1.5. 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.

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