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

What does this mean for k = 1? Does this tell us anything? There is a general inequality about any random variable X, called Chebyshev's inequality. Suppose X has mean Î¼ and standard deviation Ïƒ. Thenfor any value of k.
What does this mean for k = 2? How much of the probability must lie within two standard deviations of the mean?There is a general inequality about any random variable X, called Chebyshev's inequality. Suppose X has mean Î¼ and standard deviation Ïƒ. Thenfor any value of k.
Consider again the salaries presented in Section 6.8, Exercises 27 and 28.Income Probability20,000 ......0.4830,000 ......0.0435,000 ......0.1650,000 ......0.1257,000 ......0.04100,000 ......0.08150,000 ......0.04top salary ...0.04For each, find MAD, the variance, and coefficient of
Experiment c. For the given data (first presented in Section 6.6, Exercises 1-4), find the range, MAD, the variance both the direct way and with the computational formula, the standard deviation, and the coefficient of variation.
Experiment a. For the data first presented in Section 6.6, Exercises 37-40, find the variance and standard deviation. How many of the values lie within two standard deviations of the mean?
Experiment b. For the data first presented in Section 6.6, Exercises 37-40, find the variance and standard deviation. How many of the values lie within two standard deviations of the mean?
Experiment c. For the data first presented in Section 6.6, Exercises 37-40, find the variance and standard deviation. How many of the values lie within two standard deviations of the mean?
Experiment d. For the data first presented in Section 6.6, Exercises 37-40, find the variance and standard deviation. How many of the values lie within two standard deviations of the mean?
Estimate the standard deviation, coefficient of variation, 2.5th percentile, and 97.5th percentile from the following figures.1.2.3.4.
A p.d.f. with the same standard deviation of 10, but with a mean of 500. Calculate the coefficient of variation. Draw bell-shaped p.d.f.'s with the above properties.
Experiment d. For the given data (first presented in Section 6.6, Exercises 1-4), find the range, MAD, the variance both the direct way and with the computational formula, the standard deviation, and the coefficient of variation.
A p.d.f. with the same standard deviation of 10, but with a mean of 5. Calculate the coefficient of variation. Draw bell-shaped p.d.f.'s with the above properties.
A p.d.f. with mean of 50 and coefficient of variation of 0.4. Calculate the standard deviation. Draw bell-shaped p.d.f.'s with the above properties.
Find the variance of the random variable N1. Suppose a population obeys N1 = R1N0 where R1 is a random variable that takes on the value 1.5 with probability 0.6 and 0.5 with probability 0.4. Suppose N0 = 1.
Find the variance of the random variable ln(N1). Suppose a population obeys N1 = R1N0 where R1 is a random variable that takes on the value 1.5 with probability 0.6 and 0.5 with probability 0.4. Suppose N0 = 1.
A Bernoulli random variable with p = 1/3. Consider the above random variables that take only two values with the given probabilities. For each, find MAD, the variance, the standard deviation, and the coefficient of variation.
A Bernoulli random variable with p = 0.9. Consider the above random variables that take only two values with the given probabilities. For each, find MAD, the variance, the standard deviation, and the coefficient of variation.
A random variable that takes the value 10 with probability 1/3 and the value of 0 with probability 2/3. Compare your answers with the answer to Exercise 5. Consider the above random variables that take only two values with the given probabilities. For each, find MAD, the variance, the standard
A random variable that takes the value 11 with probability 0.9 and the value of 10 with probability 0.1. Compare your answers with the answer to Exercise 6. Consider the above random variables that take only two values with the given probabilities. For each, find MAD, the variance, the standard
The p.d.f. of a random variable X is f(x) = 2x for 0 ≤ x ≤ 1. Find the quartiles of a random variable with the given p.d.f. Illustrate the areas on a graph of the p.d.f.
An intriguing species of worm spends its time wandering around in search of food. Two percent of the plate is covered with food. A point containing food contains worms with probability 0.1. A point without food contains worms with probability 0.04. a. Draw a diagram illustrating the events and
An experiment involves placing a single worm on a plate. After 3 days, there are no worms with probability 0.1, one worm with probability 0.35, two worms with probability 0.15, three worms with probability 0.3, and four worms with probability 0.1. a. Sketch the probability distribution. b. Sketch
Baby worms grow from a length of 0.5 mm to a length of 1.0 mm. After a day of growth, the measurement L representing length has p.d.f. f (l) = 1.5(1 +l2) for 0.5 ≤ l ≤ 1.0. a. Graph the p.d.f. and check that it is consistent. b. Find the probability that a worm is less than 0.75 mm long. c.
During a 10-minute interval, adult worms switch from eating to egg-laying with probability 0.1 and from egg-laying to eating with probability 0.15. a. Draw a diagram illustrating this process. b. Derive a discrete-time dynamical system for the probability the worm is eating. c. Find the long-term
Find the marginal distributions and the distribution of Y conditional on X = 0.For the above joint distributions describing the values of the random variables X and Y, find both marginal distributions and the conditional distribution requested. Are the two random variables independent?
The random variable X has probability distribution Pr(X = 0) = 0.4 and Pr(X = 1) = 0.6. The random variable Y has probability distribution Pr(Y = 1) = 0.2 and Pr(Y = 3) = 0.8. Find the distribution of X conditional on Y = 3. Suppose the above random variables are independent. Find the joint
The random variable X has probability distribution Pr(X = 0) = 0.8 and Pr(X = 1) = 0.2. The random variable Y has probability distribution Pr(Y = 1) = 0.3, Pr(Y = 2) = 0.5, and Pr(Y = 3) = 0.2. Find the distribution of Y conditional on X = 0. Suppose the above random variables are independent. Find
The random variable X has probability distribution Pr(X = 0) = 0.3, Pr(X = l) = 0.4, and Pr(X = 2) = 0.3. The random variable Y has probability distribution Pr(Y = 1) = 0.6, Pr(Y = 2) = 0.1, and Pr(Y = 3) = 0.3. Find the distribution of Y conditional on X = 2. Suppose the above random variables are
Suppose that the random variables X and Y are each Bernoulli random variables (and thus take on only the values 0 and 1). We know that Pr(X = 0) = 0.2, Pr(Y = 0) = 0.4, and Pr(X = 0 and Y = 0) = 0.1. Use the given information to construct the entire joint distribution for the above pairs of random
Suppose that the random variables X and Y are each Bernoulli random variables and that Pr(X = 0) = 0.3, Pr(Y = 1) = 0.5, and Pr(X = 1 and Y = 0) = 0.4. Use the given information to construct the entire joint distribution for the above pairs of random variables.
Suppose that the random variables X and Y are each Bernoulli random variables, and that Pr(X = 0) = 0.3, Pr(Y = 0) = 0.6, and Pr(X = 0|Y = 0) = 0.5. Use the given information to construct the entire joint distribution for the above pairs of random variables.
Suppose that the random variables X and Y are each Bernoulli random variables, and that Pr(X = 1) = 0.8, Pr(Y = 0) = 0.4, and Pr(X = 0|Y = 1) = 0.1. Use the given information to construct the entire joint distribution for the above pairs of random variables.
Suppose measurements can only distinguish two values of T, T > 0 and T ‰¤ 0, and two values of N, N = 0 and N > 0. Find the joint distribution for these events.Consider the following joint distribution for the random variables T and N.
Suppose measurements can only distinguish two values of T, T > 0 and T ‰¤ 0, and two values of N, N ‰¤ 1 and N > 1. Find the joint distribution for these events.Consider the following joint distribution for the random variables T and N.
The case where the players have the highest possible probability of each getting a hit. When two baseball players bat in the same inning, the first gets a hit 25% of the time and the second gets a hit 35% of the time. In each of the following cases, find the joint distribution, the conditional
Find the marginal distributions and the distribution of X conditional on Y = 1For the above joint distributions describing the values of the random variables X and Y, find both marginal distributions and the conditional distribution requested. Are the two random variables independent?
The case where the players have the lowest possible probability of each getting a hit.When two baseball players bat in the same inning, the first gets a hit 25% of the time and the second gets a hit 35% of the time. In each of the following cases, find the joint distribution, the conditional
The mutants described in Section 6.2, Exercise 27, where a 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 the wild type.Write the joint distribution describing the states of the following Markov chains at times t and t + 1. Assume
The lemmings described in Section 6.2, Exercise 28 and Section 6.5, Exercise 16, where a lemming has a probability 0.2 of jumping off the cliff each hour and a probability 0.1 of crawling back up.Write the joint distribution describing the states of the following Markov chains at times t and t + 1.
Bird C. Find the conditional distributions for the number of lice on birds with zero, one, and two mites for the following birds. Describe how the conditional distributions differ from each other.
Bird D. Find the conditional distributions for the number of lice on birds with zero, one, and two mites for the following birds. Describe how the conditional distributions differ from each other.
Bird C, from Exercise 23.Draw the conditional distribution for the number of mites on birds with zero, one, and two lice for the bird given in the earlier problem.
Bird D, from Exercise 24.Draw the conditional distribution for the number of mites on birds with zero, one, and two lice for the bird given in the earlier problem.
The ecologist sees an eagle with probability 0.2 during an hour of observation, a jackrabbit with probability 0.5, and both with probability 0.05. Recall the ecologist observing eagles and rabbits. In each of the following cases, find the joint distribution, the marginal distributions, and the
The ecologist sees an eagle with probability 0.2 during an hour of observation, a jackrabbit with probability 0.5, and both with probability 0.15.Recall the ecologist observing eagles and rabbits. In each of the following cases, find the joint distribution, the marginal distributions, and the
Pr(D) = 0.2, Pr(N) = 0.8, Pr(P | D) = 1.00 and Pr(P | N) = 0.05. Find the joint distribution of the two events in the rare disease model where a person either has the disease (event D) or does not (event N) and either tests positive (event P) or does not (event Pc) in the above case. Use the joint
Find the marginal distributions and the distribution of Y conditional on X = 1.For the above joint distributions describing the values of the random variables X and Y, find both marginal distributions and the conditional distribution requested. Are the two random variables independent?
Pr(D) = 0.8, Pr(P | D) = 1.0, and Pr(P | N) = 0.1. Find the joint distribution of the two events in the rare disease model where a person either has the disease (event D) or does not (event N) and either tests positive (event P) or does not (event Pc) in the above case. Use the joint distribution
Pr(D) = 0.2, Pr(N) = 0.8, Pr(P | D) = 0.95, and Pr(P | N) = 0.05. Find the joint distribution of the two events in the rare disease model where a person either has the disease (event D) or does not (event N) and either tests positive (event P) or does not (event Pc) in the above case. Use the joint
Pr(D) = 0.8, Pr(P | D) = 0.95, and Pr(P | N) = 0.1. Find the joint distribution of the two events in the rare disease model where a person either has the disease (event D) or does not (event N) and either tests positive (event P) or does not (event Pc) in the above case. Use the joint distribution
Compare the conditional distributions with the marginal distribution with the probabilities as given. Recall the cells in Section 6.4, Exercises 31 and 32. New cells stain properly with probability 0.95, 1-day-old cells stain properly with probability 0.9, 2-day-old cells stain properly with
Compare the conditional distributions with the marginal distribution if the lab finds a way to eliminate the oldest cells (those >3 days old) from its stock.Recall the cells in Section 6.4, Exercises 31 and 32. New cells stain properly with probability 0.95, 1-day-old cells stain properly with
Find the joint distribution if immigrants enter the two populations independently.Suppose immigration and emigration change the sizes of two populations with the following probabilities.Let Ia represent the change in population a and Ib the change in population b.
Fill in the rest of the joint distribution.Suppose immigration and emigration change the sizes of two populations with the following probabilities.Let Ia represent the change in population a and Ib the change in population b.
Compare a case of meiotic drive where 60% of both pollen and ovules carry the A allele independently with a case of non-independent assortment where an offspring gets an A allele from the pollen with probability 0.6 when the ovule provides an A and gets an A allele from the pollen with probability
Compare a case of meiotic drive where 70% of the pollen and 40% of the ovules carry the A allele independently with a case of non-independent assortment where an offspring gets an A allele from the pollen with probability 0.7 when the ovule provides an A and gets an A allele from the pollen with
Many matings are observed in a species of bird. Both female and male birds come in three colors: red, blue, and green. For each experiment, find the marginal distributions for both sexes and the conditional distributions of male color for red, blue, and green females, respectively. What might be
Find the marginal distributions and the distribution of Y conditional on X = 2.For the above joint distributions describing the values of the random variables X and Y, find both marginal distributions and the conditional distribution requested. Are the two random variables independent?
Many matings are observed in a species of bird. Both female and male birds come in three colors: red, blue, and green. For each experiment, find the marginal distributions for both sexes and the conditional distributions of male color for red, blue, and green females, respectively. What might be
Find the expectations of the random variables from their marginal distributions.1. The random variables X and Y in Exercise 1.2. The random variables X and Y in Exercise 2.3. The random variables X and Y in Exercise 3.4. The random variables X and Y in Exercise 4.
The random variable X has probability distribution Pr(X = 0) = 0.7 and Pr(X = 1) = 0.3. The random variable Y has probability distribution Pr(Y = 0) = 0.3 and Pr(Y = 1) = 0.7. Find the distribution of X conditional on Y = 0. Suppose the above random variables are independent. Find the joint
(From Section 7.1, Exercises 1 and 5)For the following joint distributions, find the covariance of X and Y using the direct method, If the covariance is zero, are the random variables independent?
The case in Exercise 2.For the following joint distributions, find the correlation of X and Y.
The case in Exercise 3.For the following joint distributions, find the correlation of X and Y.
The case where the players have the highest possible probability of both getting a hit. When two baseball players bat in the same inning, the first gets a hit 25% of the time and the second gets a hit 35% of the time. In each of the following cases, find the covariance, assuming that a hit is worth
The case where the players have the lowest possible probability of both getting a hit. When two baseball players bat in the same inning, the first gets a hit 25% of the time and the second gets a hit 35% of the time. In each of the following cases, find the covariance, assuming that a hit is worth
The mutants described in Section 7.1, Exercise 21, where a gene has a 1.0% chance of mutating each time a cell divides and a 1.0% chance of correcting the mutation. In most Markov chains, the state of the system is correlated from step to step. Using the joint distribution of the states of the
The lemmings described in Section 7.1, Exercise 22, where a lemming has a probability 0.2 of jumping off the cliff each hour and a probability 0.1 of crawling back up. In most Markov chains, the state of the system is correlated from step to step. Using the joint distribution of the states of the
Y = 3X + 1. Suppose the random variable X takes on the values 0, 1, and 2 with probabilities 0.2, 0.3, and 0.5, respectively. For each of the following random variables Y, find the joint distribution of X and Y, compute the covariance and the correlation, and sketch a graph showing the
Y = X2. Suppose the random variable X takes on the values 0, 1, and 2 with probabilities 0.2, 0.3, and 0.5, respectively. For each of the following random variables Y, find the joint distribution of X and Y, compute the covariance and the correlation, and sketch a graph showing the relationship,
Y = (X - 1)2. Suppose the random variable X takes on the values 0, 1, and 2 with probabilities 0.2, 0.3, and 0.5, respectively. For each of the following random variables Y, find the joint distribution of X and Y, compute the covariance and the correlation, and sketch a graph showing the
(From Section 7.1, Exercises 2 and 6)For the following joint distributions, find the covariance of X and Y using the direct method, If the covariance is zero, are the random variables independent?
Y takes on the value 2 with probability 1. Show that this is the case with a = 0 as in Theorem 7.3. What happens when you try to compute the correlation? Explain why (think of correlation as information). Suppose the random variable X takes on the values 0, 1, and 2 with probabilities 0.2, 0.3, and
Bird C from Section 7.1, Exercise 23.Find the covariance of the number of lice and mites for the following birds from the earlier problem.
Bird D from Section 7.1, Exercise 23.Find the covariance of the number of lice and mites for the following birds from the earlier problem.
Bird C from Exercise 21.Find the correlation of the number of lice and mites for the above bird.
Bird D from Exercise 22.Find the correlation of the number of lice and mites for the above bird.
She sees an eagle with probability 0.2 during an hour of observation, a jack-rabbit with probability 0.5, and both with probability 0.05. Recall the ecologist observing eagles and rabbits in Exercises 27 and 28. Find the correlation between the random variables E and J where E = 0 represents seeing
She sees an eagle with probability 0.2 during an hour of observation, a jack-rabbit with probability 0.5, and both with probability 0.15. Recall the ecologist observing eagles and rabbits in Exercises 27 and 28. Find the correlation between the random variables E and J where E = 0 represents seeing
With the probabilities as given (as in Section 7.1, Exercise 33). Recall the cells in Section 6.4, Exercises 31 and 32. New cells stain properly with probability 0.95, 1-day-old cells stain properly with probability 0.9, 2-day-old cells stain properly with probability 0.8, and 3-day-old cells stain
If the lab finds a way to eliminate the oldest cells (>3 days old) from its stock (as in Exercise 34). Recall the cells in Section 6.4, Exercises 31 and 32. New cells stain properly with probability 0.95, 1-day-old cells stain properly with probability 0.9, 2-day-old cells stain properly with
Suppose that the bird gives up eating (as in the text) and spends 10 h sleeping when the weather is nice, 14 h sleeping when the weather is OK, and 18 h sleeping when the weather is bad. Find pS,P.Consider birds that spend all of their time sleeping, eating, or preening. Let S be the time spent
(From Section 7.1, Exercises 3 and 7)For the following joint distributions, find the covariance of X and Y using the direct method, If the covariance is zero, are the random variables independent?
The bird always eats for 4 h per day and spends 10 h sleeping when the weather is nice, 14 h sleeping when the weather is OK, and 18 h sleeping when the weather is bad. Find pS,P. Why are pE,P and pE,S not worth finding?Consider birds that spend all of their time sleeping, eating, or preening. Let
When the weather is nice, the bird eats for 1 h and sleeps for 10 h. When the weather is OK, the bird eats for 1 h and sleeps for 14 h. When the weather is bad, the bird eats for 6 h and sleeps for 18 h. Find pS,P. Why is the correlation not perfect?Consider birds that spend all of their time
When the weather is nice, the bird eats for 6 h and sleeps for 10 h. When the weather is OK, the bird eats for 1 h and sleeps for 14 h. When the weather is bad, the bird eats for 1 h and sleeps for 18 h. Find pS,P. Why is the correlation not perfect?Consider birds that spend all of their time
Explicitly compute the covariance if immigrants enter the two populations independently.Suppose immigration and emigration change the sizes of two populations with the following probabilities.Let Ia represent the change in population a and Ib the change in population b.
Compute the covariance in the case from Section 7.1, Exercise 36.Suppose immigration and emigration change the sizes of two populations with the following probabilities.Let Ia represent the change in population a and Ib the change in population b.
Suppose the following are measurements of the temperature T and insect size S.1. Find the correlation of S with T.2. Find the correlation of ln(S) with T.
Consider the following data for cell age A and the number of toxic molecules N inside.1. Find the correlation of A with N.2. Suppose that the damage done by the toxic molecules is D = ln(1 + N). Find the correlation of A with D.
(From Section 7.1, Exercises 4 and 8)For the following joint distributions, find the covariance of X and Y using the direct method,If the covariance is zero, are the random variables independent?
The case in Exercise 1.For the following joint distributions, find the covariance of X and Y using the computational method, Cov(X, Y) = E(XY) - .
The case in Exercise 2.For the following joint distributions, find the covariance of X and Y using the computational method, Cov(X, Y) = E(XY) - .
The case in Exercise 3.For the following joint distributions, find the covariance of X and Y using the computational method, Cov(X, Y) = E(XY) - .
The case in Exercise 4.For the following joint distributions, find the covariance of X and Y using the computational method, Cov(X, Y) = E(XY) - .
(From Section 7.1, Exercises 1 and 5)For the following joint distributions, find the probabilities for the random variable X + Y and check that E(X + Y) = E(X) + E(Y).
The random variables X and Y in Exercise 2, with variances found in Section 7.2, Exercise 10.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.
The random variables X and Y in Exercise 3, with variances found in Section 7.2, Exercise 11.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.
The random variables X and Y in Exercise 4.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.

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