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

Find Cov(X, X). How does it compare with Var(X)? Consider any random variable X that has a finite expectation and variance.
Consider the new random variable Y = -X. Find Cov(X, Y). How does it compare with Var(X)? Consider any random variable X that has a finite expectation and variance.
X and Y take the value 0 with probability 0.5 and the value 1 with probability 0.5.Consider independent random variables X and Y with identical probability distributions as given. Let Z = X + X and S = X + Y.a. Compute the mean and variance of Z directly from its probability distribution.b. Compute
X and Y take the value 0 with probability 0.25, the value 1 with probability 0.5, and the value 2 with probability 0.25.Consider independent random variables X and Y with identical probability distributions as given. Let Z = X + X and S = X + Y.a. Compute the mean and variance of Z directly from
Use the following steps to prove that the geometric mean of the product of two random variables X and Y is equal to the product of the geometric means.1. Write the definition of the geometric mean of the product XY.2. Use a law of logs to expand the result.3. Use Theorem 7.4 to break up the
(From Section 7.1, Exercises 2 and 6)For the following joint distributions, find the probabilities for the random variable X + Y and check that E(X + Y) = E(X) + E(Y).
X takes the value 1 with probability 0.5 and the value 2 with probability 0.5. The harmonic mean is another kind of average (like the geometric mean), defined by H(X) = 1/E(1/X) (the reciprocal of the expectation of the reciprocal). Like the geometric mean, the harmonic mean is only defined for
Suppose X takes the value 1 with probability 0.1 and the value 10 with probability 0.9. The harmonic mean is another kind of average (like the geometric mean), defined by H(X) = 1/E(1/X) (the reciprocal of the expectation of the reciprocal). Like the geometric mean, the harmonic mean is only
Suppose X and Y are independent random variables that each take the value 1 with probability 0.5 and the value 2 with probability 0.5. Use the following random variables to show that E(X/Y) ≠ E(X)/E(Y) (the expectation of the quotient is not equal to the quotient of the expectations) even when
Suppose X and Y are independent random variables where X takes the value 0 with probability 0.5 and the value 1 with probability 0.5, and Y takes the value 1 with probability 0.1 and the value 10 with probability 0.9. Use the following random variables to show that E(X/Y) ≠ E(X)/E(Y) (the
Bird C. Consider again the birds suffering from mites and lice in Section 7.1, Exercises 23 and 24. Find the probability distribution of P = L + M and use it to compute E(P) directly. Compare the result with Theorem 7.4.
Bird D. Consider again the birds suffering from mites and lice in Section 7.1, Exercises 23 and 24. Find the probability distribution of P = L + M and use it to compute E(P) directly. Compare the result with Theorem 7.4.
Bird C. Consider yet again the birds suffering from mites and lice. For each bird, find Var(P) directly from the probability distribution of P = L + M. Show how you could have found the variance with the general addition formula for variances.
Bird D. Consider yet again the birds suffering from mites and lice. For each bird, find Var(P) directly from the probability distribution of P = L + M. Show how you could have found the variance with the general addition formula for variances.
Bird A (from Example 7.3.5).Find the probability distribution of W = 0.05L + 0.02M for the following birds. Check that the expectation is 0.059 in both cases.
(From Section 7.1, Exercises 3 and 7)For the following joint distributions, find the probabilities for the random variable X + Y and check that E(X + Y) = E(X) + E(Y).
Bird B (from Table 7.1). Find the probability distribution of W = 0.05L + 0.02M for the following birds. Check that the expectation is 0.059 in both cases.
Find the expected number of each type of immigrant. Find the expected total number of immigrants from all species combined.Suppose annual immigration into a park by three species follows the probabilities in the table.
Suppose species 1 in has mass 10 kg, species 2 has mass 5 kg, and species 3 has mass 15 kg. Find the expected mass of the immigrants of each species that arrive. Find the expected total mass of all immigrants.Suppose annual immigration into a park by three species follows the probabilities in the
Ignore the third immigrant, species and suppose that species 1 and species 2 arrive independently.a. Give the joint probability distribution for species 1 and species 2.b. Find the probability of each possible number of immigrants.c. Find the expected number of immigrants of each of these two
The first two immigrant species are appealing to eco-tourists, and each additional individual brings in $1000. The third species is repellent to eco-tourists and reduces revenue by $500.a. Find the expected revenue from each species separately and for all species together.b. Find the variance in
Consider the situation described in Exercise 35 but suppose that the two species do not arrive independently. Find a set of probabilities for outcomes consistent with the probabilities that you think will have higher variance than the independent case. Compute the expectation and variance.Suppose
Consider the situation described in Exercise 35 but suppose that the two species do not arrive independently. Find a set of probabilities for outcomes consistent with the probabilities that you think will have lower variance than the independent case. Compute the expectation and variance.Suppose
(From Section 7.1, Exercises 4 and 8)For the following joint distributions, find the probabilities for the random variable X + Y and check that E(X + Y) = E(X) + E(Y).
Find the expected total calories per caterpillar for each bird.Consider the following table giving the probability that three types of birds consume two different species of caterpillars that come in two different sizes. The first has food quality (in kCal/cm3) of 1.0, and the second has food
From the expected total calories for each bird, and find the covariance of quality and volume.Consider the following table giving the probability that three types of birds consume two different species of caterpillars that come in two different sizes. The first has food quality (in kCal/cm3) of
Suppose that the probability that a cell is cancerous is AN/10.Consider the following data for cell age A and the number of toxic molecules N inside.Suppose that the probability that a cell is cancerous depends on both the age and the number of toxic molecules. Use the law of total probability to
Suppose that the probability that a cell is cancerous is AN2/20.Consider the following data for cell age A and the number of toxic molecules N inside.Suppose that the probability that a cell is cancerous depends on both the age and the number of toxic molecules. Use the law of total probability to
The random variables X and Y in Exercise 1, with covariance found in Section 7.2, Exercise 1. For the following joint distributions, find the probabilities for the random variable XY (the product) and check that E(XY) = E(X)E(Y) only if Cov(X, Y) = 0.
The random variables X and Y in Exercise 2, with covariance found in Section 7.2, Exercise 2. For the following joint distributions, find the probabilities for the random variable XY (the product) and check that E(XY) = E(X)E(Y) only if Cov(X, Y) = 0.
The random variables X and Y in Exercise 3, with covariance found in Section 7.2, Exercise 3. For the following joint distributions, find the probabilities for the random variable XY (the product) and check that E(XY) = E(X)E(Y) only if Cov(X, Y) = 0
The random variables X and Y in Exercise 4, with covariance found in Section 7.2, Exercise 4. For the following joint distributions, find the probabilities for the random variable XY (the product) and check that E(XY) = E(X)E(Y) only if Cov(X, Y) = 0.
The random variables X and Y in Exercise 1.For the following joint distributions, find the probabilities for the random variable X - Y (the difference), and check that E(X - Y) = E(X) - E(Y) and that Var(X - Y) = Var(X) + Var(Y) if Cov(X, Y) = 0.
Evaluate the following. Find all the factorials explicitly.1.2.3.4.5.6.
Calculate the given value. b(2; 6, 0.4) (Use the value computed in Exercise 4.)
Calculate the given value. b(1; 7, 0.2) (Use the value computed in Exercise 5.)
Calculate the given value. b(2; 7, 0.6) (Use the value computed in Exercise 6.)
n = 4, p = 0.3. Find the expectation, variance, and mode for binomial random variables with the above parameter.
n = 6, p = 0.4. Find the expectation, variance, and mode for binomial random variables with the above parameter.
n = 7, p = 0.1. Find the expectation, variance, and mode for binomial random variables with the above parameter.
n = 17, p = 0.6. Find the expectation, variance, and mode for binomial random variables with the above parameter.
A binomial random variable B with p = 0.7 and n = 2. Use the formula for the binomial probability distribution to find and graph the probability distribution in the above case.
A binomial random variable B with p = 0.4 and n = 2. Use the formula for the binomial probability distribution to find and graph the probability distribution in the above case.
A binomial random variable B with p = 0.7 and n = 3. Use the formula for the binomial probability distribution to find and graph the probability distribution in the above case.
A binomial random variable B with p = 0.4 and n = 3. Use the formula for the binomial probability distribution to find and graph the probability distribution in the above case.
A binomial random variable B with p = 0.7 and n = 2. Compute the mean and the variance from the probability distribution and make sure that your answers match the formulas in Equations 7.4.2 and 7.4.3.
A binomial random variable B with p = 0.4 and n = 2. Compute the mean and the variance from the probability distribution and make sure that your answers match the formulas in Equations 7.4.2 and 7.4.3.
A binomial random variable B with p = 0.7 and n = 3. Compute the mean and the variance from the probability distribution and make sure that your answers match the formulas in Equations 7.4.2 and 7.4.3.
A binomial random variable B with p = 0.4 and n = 3. Compute the mean and the variance from the probability distribution and make sure that your answers match the formulas in Equations 7.4.2 and 7.4.3.
List all ways to get three successes out of four trials, and find the probability of each outcome. Suppose the probability of a success is p. Find the probability of each of the above event and compare with the formula for the binomial distribution.
List all ways to get two successes out of four trials, and find the probability of each outcome. Suppose the probability of a success is p. Find the probability of each of the above event and compare with the formula for the binomial distribution.
Suppose two trials are independent, but the first has a probability 0.3 of success, and the second a probability 0.7 of success. Find the probabilities of zero, one, and two successes, and compare with the binomial distribution with n = 2 and p = 0.5 (the average of the two probabilities). Does the
Suppose two trials are independent, but the first has a probability 0.3 of success, and the second a probability 0.1 of success. Find the probabilities of zero, one, and two successes, and compare with the binomial distribution with n = 2 and p = 0.2. Does the expected number of successes match the
Suppose the first trial has probability of success 0.5, and the second is successful with probability 0.8 if the first is and succeeds with probability 0.2 if the first fails. Show that the second trial has a probability 0.5 of success. Find the probabilities of zero, one, and two successes, and
Suppose the first trial has probability of success 0.2, and the second is successful with probability 0 if the first is and succeeds with probability 0.25 if the first fails. Show that the second trial has a probability 0.2 of success. Find the probabilities of zero, one, and two successes, and
Show thatExplain why this must be true.The valueshave many beautiful mathematical properties. Here are just a few.
Figure out why the following induction should hold, and show that it does.The valueshave many beautiful mathematical properties. Here are just a few.
The values ofare also called binomial coefficients. Expand (x + l)3 and (x + l)4 and check that the coefficient of the kth power of x isin the first case andin the second.The valueshave many beautiful mathematical properties. Here are just a few.
Explain why the coefficients of the powers of x in the expansion of (x + 1)n are the binomial coefficients. What is the connection with Exercise 32?The valueshave many beautiful mathematical properties. Here are just a few.
Family C has eight children, seven of whom are girls. Family D also has eight children, four of whom are girls. Which type of family is more probable? Suppose that the probability that a baby is a boy is 0.5 and that a baby is a girl is also 0.5. Find the probabilities of each of the following
Family C has eight children: three girls, one boy, and then four more girls. Family D also has eight children: two girls, one boy, one girl, three boys, and then a girl. Which type of family is more probable? Suppose that the probability that a baby is a boy is 0.5 and that a baby is a girl is also
A group of identical quintuplets named Aaron, Bill, Carl, Dave, and Ed enjoy confusing the teachers at their school. List the number of different possibilities in each case, and then count them up. Make sure your counts match the appropriate value of "n choose k."1. Only one goes to school.2. Two
Find the number of ways the following can be ordered. List three of the possible orderings.1. The order of finishing by three horses (named Speedy, Blinky, and Sparky) in a race.2. The items in a four-course meal (soup, salad, main dish, and dessert).3. A five-card poker hand (ace, 2, 5, 10,
What is the probability that the second patient gets the right medication conditional on the first getting the right medication? Are the two events independent? Each of five patients has been prescribed a different medication, but the prescriptions were accidentally shuffled. Compute the above
What is the probability that the second patient gets the wrong medication conditional on the first getting the wrong medication? Are the two events independent? Each of five patients has been prescribed a different medication, but the prescriptions were accidentally shuffled. Compute the above
Suppose that three of the patients were prescribed one medication and the other two were prescribed a different one. What is the probability that all five get the right medication? Each of five patients has been prescribed a different medication, but the prescriptions were accidentally shuffled.
Find the probability distribution for the number of heterozygous offspring. Find the expectation and variance. Sketch the distribution. Suppose a heterozygous plant self-pollinates and produces five offspring with independent genotypes.
Suppose that one allele is dominant and produces tall plants. Find and sketch the probability distribution for the number of tall offspring. Find the expectation and variance of the number of tall offspring. Suppose a heterozygous plant self-pollinates and produces five offspring with independent
Suppose that all of the homozygous offspring survive and half of the heterozygous offspring survive. We have seen that non independence of alleles (possibly caused by differential mortality of genotypes) can lead to deviations from normal proportions of offspring genotypes. Find the probability
Suppose that all offspring with genotype AA survive, half of the offspring with genotype Aa survive, and one fourth of the offspring with genotype aa survive. We have seen that non independence of alleles (possibly caused by differential mortality of genotypes) can lead to deviations from normal
Suppose meiotic drive affects the pollen only and that 80% of the pollen grains from a heterozygote carry the A allele. Ovules are normal and 50% of them carry the A allele. We have seen that meiotic drive (where one allele pushes its way into more than half of the gametes) can lead to deviations
Suppose meiotic drive affects both pollen and ovules and that 80% of the pollen grains and ovules from a heterozygote carry the A allele. We have seen that meiotic drive (where one allele pushes its way into more than half of the gametes) can lead to deviations from normal proportions of offspring
b(1; 4, 0.3) (Use the value computed in Exercise 1.) Calculate the given value.
Calculate the given value. b(3; 4, 0.3) (Use the value computed in Exercise 2.)
Calculate the given value. b(2; 5, 0.4) (Use the value computed in Exercise 3.)
Exactly three out of four 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
Out of three offspring, one is tall, and two are intermediate. 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 Aa individuals
Out of four offspring, two are tall, and two are intermediate. 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 Aa individuals
Out of four offspring, one is tall, two are 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
Out of three offspring, one is tall, one is intermediate, and one is short.There is a formula for the multinomial distribution describing probabilities when there are more than two outcomes of a trial. Suppose there are three possible outcomes of each trial, numbered 1 through 3, with probabilities
Out of three offspring, one is tall, and two are intermediate.There is a formula for the multinomial distribution describing probabilities when there are more than two outcomes of a trial. Suppose there are three possible outcomes of each trial, numbered 1 through 3, with probabilities p1, p2, and
Out of four offspring, two are tall, and two are intermediate.There is a formula for the multinomial distribution describing probabilities when there are more than two outcomes of a trial. Suppose there are three possible outcomes of each trial, numbered 1 through 3, with probabilities p1, p2, and
Out of four offspring, one is tall, two are intermediate, and one is short.There is a formula for the multinomial distribution describing probabilities when there are more than two outcomes of a trial. Suppose there are three possible outcomes of each trial, numbered 1 through 3, with probabilities
Find the expectation and the mode of the number of tall offspring. Suppose that the alleles A and a for height are additive, meaning that plants with genotype AA are tall, plants with genotype Aa are intermediate, and those with genotype aa are short. If an Aa plant is crossed with another Aa
Find the expectation and the mode of the number of intermediate offspring. Suppose that the alleles A and a for height are additive, meaning that plants with genotype AA are tall, plants with genotype Aa are intermediate, and those with genotype aa are short. If an Aa plant is crossed with another
Find the probability that the number of tall offspring is less than or equal to the mode.Suppose that the alleles A and a for height are additive, meaning that plants with genotype AA are tall, plants with genotype Aa are intermediate, and those with genotype aa are short. If an Aa plant is crossed
Exactly six 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
Find the probability that the number of intermediate offspring is less than or equal to the mode.Suppose that the alleles A and a for height are additive, meaning that plants with genotype AA are tall, plants with genotype Aa are intermediate, and those with genotype aa are short. If an Aa plant is
Find the mean number of islands occupied. Consider two sets of ten islands. In the first set, each island has a 0.2 chance of switching from empty to occupied, and a 0.1 chance of switching from occupied to empty. The equilibrium fraction occupied is 2/3. In the second set of ten islands, all are
Find the variance of the number of islands occupied. Consider two sets of ten islands. In the first set, each island has a 0.2 chance of switching from empty to occupied, and a 0.1 chance of switching from occupied to empty. The equilibrium fraction occupied is 2/3. In the second set of ten
Find the mode of the number of islands occupied. Consider two sets of ten islands. In the first set, each island has a 0.2 chance of switching from empty to occupied, and a 0.1 chance of switching from occupied to empty. The equilibrium fraction occupied is 2/3. In the second set of ten islands,
Sketch the probability distribution for each set of islands. Why does the second set fail to follow the binomial distribution? Consider two sets of ten islands. In the first set, each island has a 0.2 chance of switching from empty to occupied, and a 0.1 chance of switching from occupied to empty.
Consider the mutant genes described in Section 6.2, Exercise 27, where a wild type gene has a 1.0% chance of mutating each time a cell divides and a mutant gene has a 1.0% chance of reverting to wild type. Suppose that four genes start out normal. Find the probability that there are two or more
Consider the lemmings described in Section 6.2, Exercise 28, where a lemming has a probability 0.2 of jumping off the cliff each hour and a probability 0.1 of crawling back up. Suppose that five lemmings start at top. Find the probability that more than half of the lemmings are at the bottom after
1 min. Starting with five molecules, each leaving with probability 0.2 per minute, compute and graph the probability distribution describing the number remaining at the following times.
One or fewer out of four 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
2 min. Starting with five molecules, each leaving with probability 0.2 per minute, compute and graph the probability distribution describing the number remaining at the following times.
5 min. Starting with five molecules, each leaving with probability 0.2 per minute, compute and graph the probability distribution describing the number remaining at the following times.
10 min. Starting with five molecules, each leaving with probability 0.2 per minute, compute and graph the probability distribution describing the number remaining at the following times.
Exactly one remains. 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.

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