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

The histogram in Exercise 4. Find and sketch the cumulative distribution associated with the histogram from the earlier problem.
On each histogram, find the most and least likely simple events. Is the histogram symmetric?1.2.3.4.
Experiment a. For the data presented in Section 6.6, Exercises 1-4, write the results in terms of a random variable and find the expectation. What fraction of experiments have a result less than the expectation?
The p.d.f. of a random variable T is g(t) = 6t(1 - t) for 0 ≤ t ≤ 1. Find the expectation of the continuous random variables with the given p.d.f. Find the probability that the random variable has a value less than the expectation.
The p.d.f. of a random variable X is f (x) = 2x for 0 ≤ x ≤ 1. Suppose that measurements are very imprecise, and that all values of X ≤ 0.5 are recorded as 0.25 and all values of X > 0.5 are recorded as 0.75. Write a random variable describing these imprecise measurements, find the associated
The p.d.f. of a random variable X is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. Suppose that measurements are very imprecise, and that all values of X ≤ 1 are recorded as 0.5 and all values of X > 1 are recorded as 1.5. Write a random variable describing these imprecise measurements, find the associated
The p.d.f. of a random variable X is f(x) = 2x for 0 ≤ x ≤ 1. Suppose that measurements are imprecise, and that all values of X ≤ 0.25 are recorded as 0.25, all values of 0.25 < X ≤ 0.5 are recorded as 0.5, all values of 0.5 < X ≤ 0.75 are recorded as 0.75, and all values of 0.75 < X are
The p.d.f. of a random variable X is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. Suppose that measurements are imprecise, and that all values of X ≤ 0.5 are recorded as 0.25, all values of 0.5 < X ≤ 1.0 are recorded as 0.75, all values of 1.0 < X ≤ 1.5 are recorded as 1.25, and all values of 1.5 < X
Check that the following could be p.d.f.'s and compute their expectations. Does anything seem odd about them?
Check that the following could be p.d.f.'s and compute their expectations. Does anything seem odd about them?
A Bernoulli random variable describing whether a molecule is in or out at time 1. Think about one or more molecules independently leaving a cell, each with probability 0.9 in a given second. Find the random variable describing the following events, find the probabilities of the outcomes, and find
A Bernoulli random variable describing whether a molecule is in or out at time 2. Think about one or more molecules independently leaving a cell, each with probability 0.9 in a given second. Find the random variable describing the following events, find the probabilities of the outcomes, and find
A Bernoulli random variable describing whether two molecules are together (both in or both out) or separate at time l. Think about one or more molecules independently leaving a cell, each with probability 0.9 in a given second. Find the random variable describing the following events, find the
Experiment b. For the data presented in Section 6.6, Exercises 1-4, write the results in terms of a random variable and find the expectation. What fraction of experiments have a result less than the expectation?
A Bernoulli random variable describing whether three out of three molecules remain inside at time 1. Think about one or more molecules independently leaving a cell, each with probability 0.9 in a given second. Find the random variable describing the following events, find the probabilities of the
A discrete random variable that counts the number out of two molecules that are in at time 1. Think about one or more molecules independently leaving a cell, each with probability 0.9 in a given second. Find the random variable describing the following events, find the probabilities of the
A discrete random variable that counts the number out of two molecules that are in at time 2. Think about one or more molecules independently leaving a cell, each with probability 0.9 in a given second. Find the random variable describing the following events, find the probabilities of the
Experiment a. For the data presented in Section 6.6, Exercises 37-40, write the results in terms of a random variable and find the expectation. What fraction of experiments have a result less than the expectation?
Experiment b. For the data presented in Section 6.6, Exercises 37-40, write the results in terms of a random variable and find the expectation. What fraction of experiments have a result less than the expectation?
Experiment c. For the data presented in Section 6.6, Exercises 37-40, write the results in terms of a random variable and find the expectation. What fraction of experiments have a result less than the expectation?
Experiment d. For the data presented in Section 6.6, Exercises 37-40, write the results in terms of a random variable and find the expectation. What fraction of experiments have a result less than the expectation?
Suppose 30% of the cells are young. Consider again the cells in Section 6.4, Exercises 29 and 30, but suppose that older cells, instead of not staining as often, do not stain as well. In particular, they produce a brightness of 7, while young cells have a brightness of 9. Write a random variable
Suppose 70% of the cells are young. Consider again the cells in Section 6.4, Exercises 29 and 30, but suppose that older cells, instead of not staining as often, do not stain as well. In particular, they produce a brightness of 7, while young cells have a brightness of 9. Write a random variable
Suppose Pr(cell is 0 day old) = 0.4 Pr(cell is 1 day old) = 0.3 Pr(cell is 2 days old) = 0.2 Pr(cell is 3 days old) = 0.1. Consider again the cells in Section 6.4, Exercises 31 and 32, but suppose that older cells, instead of not staining as often, do not stain as well. New cells have a brightness
Experiment c. For the data presented in Section 6.6, Exercises 1-4, write the results in terms of a random variable and find the expectation. What fraction of experiments have a result less than the expectation?
The lab finds a way to eliminate the oldest cells (more than 3 days old) from its stock. Consider again the cells in Section 6.4, Exercises 31 and 32, but suppose that older cells, instead of not staining as often, do not stain as well. New cells have a brightness of 9.5, 1-day-old cells have a
Population a. How many immigrants do you think would arrive (or depart) in 10 years? Will the population grow?Suppose immigration and emigration change the sizes of four populations with the following probabilities.Write the result as a random variable and find the expectation.
Population b. How many immigrants do you think would arrive (or depart) in 10 years? Will the population grow?Suppose immigration and emigration change the sizes of four populations with the following probabilities.Write the result as a random variable and find the expectation.
Population c. How many immigrants would arrive (or depart) in 10 years? Will the population grow? Does the expectation seem close to the "middle" of the distribution?Suppose immigration and emigration change the sizes of four populations with the following probabilities.Write the result as a random
Population d. About how many immigrants would arrive (or depart) in 10 years? Will the population grow? Does the expectation seem close to the "middle" of the distribution?Suppose immigration and emigration change the sizes of four populations with the following probabilities.Write the result as a
As in Section 6.6, Exercises 45 and 46, the p.d.f. for the waiting time X until an event occurs often follows the exponential distribution, with the form g(x) = Î»e-Î»x for some positive value of Î», defined for x ‰¥ 0. Use the following indefinite integral fact to find the expectation for
A 20 kg child is the end of a long see saw, 3 m to the right of center. An 80 kg adult is 1 m to the left of center. Find the center of mass. Who will go up?The formula for the expectation in mathematics is identical to the formula for the center of mass in physics. Mass acts like probability, and
A 20 kg child is 3 m to the right of center, her 30 kg older brother is 1 m to the right of center, and an 80 kg adult is 1 m to the left of center. Find the center of mass. Who will go up?The formula for the expectation in mathematics is identical to the formula for the center of mass in physics.
A 20 kg child is 3 m to the right of center. An 80 kg adult wishes to balance the see saw by sitting a distance x to the left of center. They will balance if the center of mass is exactly 0. Solve for x.The formula for the expectation in mathematics is identical to the formula for the center of
Experiment d. For the data presented in Section 6.6, Exercises 1-4, write the results in terms of a random variable and find the expectation. What fraction of experiments have a result less than the expectation?
A 20 kg child is 3 m to the right of center, her 30 kg older brother is 1 m to the right of center, and an 80-kg adult wishes to balance the see saw by sitting a distance x to the left of center. Solve for x.The formula for the expectation in mathematics is identical to the formula for the center
Find the center of mass of a bar with mass density p(x) = x(l - x2) for 0 ‰¤ x ‰¤ 1. Is it to the left or the right of the center of the bar at x = 1 /2? Is it at the point where p(x) takes on its maximum? Sketch a graph.The relation between the mathematical expectation and the center of mass
Find the center of mass of a bar with mass density p(x) = x2(2 - x) for 0 ‰¤ x ‰¤ 2. Is it to the left or the right of the center of the bar at x = 1 ? Is it at the point where p(x) takes on its maximum?The relation between the mathematical expectation and the center of mass in physics also
As in Exercise 37, a 4 m long wooden board has a fulcrum placed 1 m from the left end. A 20 kg child sits at the right end of the board (at position x = 3) and an 80 kg adult sits at the left end (at position x = -1). Suppose that the board has density of 5 kg/m. Find the center of mass. Who will
As in Exercise 37, a 4 m long wooden board has a fulcrum placed 1 m from the left end. A 20 kg child sits at the right end of the board (at position x = 3), and an 80 kg adult sits at the left end (at position x = -1). Suppose that the board has density of y kg/m. Find the value of y for which the
At time 4. Find the expected number of molecules in the cell using the data in Example 6.7.19 at the following times.
At time 8. Find the expected number of molecules in the cell using the data in Example 6.7.19 at the following times.
The p.d.f. of a random variable X is f(x) = 2x for 0 ≤ x ≤ 1. Find the expectation of the continuous random variables with the given p.d.f. Find the probability that the random variable has a value less than the expectation.
The p.d.f. of a random variable X is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. Find the expectation of the continuous random variables with the given p.d.f. Find the probability that the random variable has a value less than the expectation.
The p.d.f. of a random variable T is h(t) = 1/t for 1 ≤ t ≤ e. Find the expectation of the continuous random variables with the given p.d.f. Find the probability that the random variable has a value less than the expectation.
Experiment a. Consider again the data presented in Section 6.6, Exercises 1-4,Find the median and the mode, and compare with the expectation. When is the median greater than the expectation?
The p.d.f. of a random variable X is f (x) = 1 - x/2 for 0 ≤ x ≤ 2. Find the median and mode of the continuous random variables with the given p.d.f., and compare with the expectation as found earlier.
The p.d.f. of a random variable T is h(t) = 1/t for 1 ≤ t ≤ e. Find the median and mode of the continuous random variables with the given p.d.f., and compare with the expectation as found earlier.
The p.d.f. of a random variable T is g(t) = 6t(l - t) for 0 ≤ t ≤ 1. Find the median and mode of the continuous random variables with the given p.d.f., and compare with the expectation as found earlier.
Recall the following slightly peculiar p.d.f.'s from Section 6.7, Exercises 15 and 16. Find the median of each. How does it compare with the expectation?1.2.
The random variable X where Pr(X = 1) = 0.3 and Pr(X = 2) = 0.7. Find the arithmetic and geometric means of the above random variables. Check that the arithmetic-geometric inequality holds.
The random variable X where Pr(X = 1) = 0.4 and Pr(X = 3) = 0.6. Find the arithmetic and geometric means of the above random variables. Check that the arithmetic-geometric inequality holds.
The random variable X where Pr(X = 1) = 0.3, Pr(X = 2) = 0.3, and Pr(X = 3) = 0.4. Find the arithmetic and geometric means of the above random variables. Check that the arithmetic-geometric inequality holds.
The random variable X where Pr(X = 2) = 0.1, Pr(X = 3) = 0.2, and Pr(X = 5) = 0.7. Find the arithmetic and geometric means of the above random variables. Check that the arithmetic-geometric inequality holds.
The p.d.f. of a random variable X is f(x) = 2 for 0.75 ≤ x ≤ 1.25. Use Equation 6.8.2. Find the geometric mean of the continuous random variables with the given p.d.f. Compare with the expectation.
Experiment b. Consider again the data presented in Section 6.6, Exercises 1-4,Find the median and the mode, and compare with the expectation. When is the median greater than the expectation?
The p.d.f. of a random variable X is f(x) = 5 for 1 ≤ x ≤ 1.2. Find the geometric mean of the continuous random variables with the given p.d.f. Compare with the expectation.
The p.d.f. of a random variable X is f(x) = 2x for 0 ≤ x ≤ 1. Use Equation 6.8.3, and you will have to use L'Hopital's rule to evaluate the integral. Find the geometric mean of the continuous random variables with the given p.d.f. Compare with the expectation.
The p.d.f. of a random variable T is h(t) = 1/t for 1 ≤ t ≤ e. Find the geometric mean of the continuous random variables with the given p.d.f. Compare with the expectation.
Check this in the case that r1 = 1 and r2 = 2. The geometry behind the geometric mean is based on the following argument. If a random variable R takes on each of the values r1 and r2 with probability 0.5, a rectangle with sides of length r1 and r2 has area equal to that of a square with sides with
Check this in general, without picking values for r1 and r2. The geometry behind the geometric mean is based on the following argument. If a random variable R takes on each of the values r1 and r2 with probability 0.5, a rectangle with sides of length r1 and r2 has area equal to that of a square
Fix r1 = 1. Find the value of r2 that maximizes the ratio of the geometric mean to the arithmetic mean. The geometry behind the geometric mean is based on the following argument. If a random variable R takes on each of the values r1 and r2 with probability 0.5, a rectangle with sides of length r1
Prove that the geometric mean is always less than or equal to the arithmetic mean (the arithmetic-geometric inequality) when r1 = 1 (the case described in Exercise 25). The geometry behind the geometric mean is based on the following argument. If a random variable R takes on each of the values r1
The top salary is $450,000. Suppose that incomes in a company have the following probabilities. Income Probability 20,000 ...... 0.48 30,000 ...... 0.04 35,000 ...... 0.16 50,000 ...... 0.12 57,000 ...... 0.04 100,000 ...... 0.08 150,000 ...... 0.04 top salary ... 0.04 For the given values of the
The top salary is $4,500,000. Suppose that incomes in a company have the following probabilities. Income Probability 20,000 ...... 0.48 30,000 ...... 0.04 35,000 ...... 0.16 50,000 ...... 0.12 57,000 ...... 0.04 100,000 ...... 0.08 150,000 ...... 0.04 top salary ... 0.04 For the given values of the
Experiment a. For the data (from Section 6.6, Exercises 37-40), find the median and the mode.
Experiment c. Consider again the data presented in Section 6.6, Exercises 1-4,Find the median and the mode, and compare with the expectation. When is the median greater than the expectation?
Experiment b. For the data (from Section 6.6, Exercises 37-40), find the median and the mode.
Experiment c. For the data (from Section 6.6, Exercises 37-40), find the median and the mode.
Experiment d. For the data (from Section 6.6, Exercises 37-40), find the median and the mode.
As in Section 6.6, Exercises 45 and 46, the p.d.f. for the waiting time X until an event occurs often follows the exponential distribution, with the form g(x) = λe-λx for some positive value of λ, defined for x ≥ 0. Find the median waiting time for the following values of ≥ and compare with
There are two cities, one with 100,000 people and the other with 1,000,000 people. The above problem shows that most people live in places that are more crowded than average. In each case, find the average size of a city, the average crowding a person experiences, and the fraction of people who
There are three cities, one with 100,000 people, one with 400,000 people, and the other with 1,000,000 people. The above problem shows that most people live in places that are more crowded than average. In each case, find the average size of a city, the average crowding a person experiences, and
R = 4 with probability 0.5, R = 0.25 with probability 0.5. Find the arithmetic and geometric means of the random variables R for per capita production in the above cases. Check that the arithmetic-geometric inequality holds in each case. Which describes a growing population?
R = 4 with probability 0.25, R = 0.25 with probability 0.75. Find the arithmetic and geometric means of the random variables R for per capita production in the above cases. Check that the arithmetic-geometric inequality holds in each case. Which describes a growing population?
R = 4 with probability 0.75, R = 0.25 with probability 0.25. Find the arithmetic and geometric means of the random variables R for per capita production in the above cases. Check that the arithmetic-geometric inequality holds in each case. Which describes a growing population?
Experiment d. Consider again the data presented in Section 6.6, Exercises 1-4,Find the median and the mode, and compare with the expectation. When is the median greater than the expectation?
R = 5 with probability 0.25, R = 0.25 with probability 0.25, R = 1 with probability 0.5. Find the arithmetic and geometric means of the random variables R for per capita production in the above cases. Check that the arithmetic-geometric inequality holds in each case. Which describes a growing
Suppose populations start at 100. Estimate the population size after 50 generations in the following cases.1. The situation in Exercise 37.2. The situation in Exercise 38.3. The situation in Exercise 39.4. The situation in Exercise 40.
In week 1, the "low price" manager cuts prices by 50%. In week 2, the "high price" manager raises prices by 50%, and so forth. Suppose an item started out at $100.A store has two managers, one who believes that high profits come from lowering prices and getting more customers and another who
In week 1, the "low price" manager cuts prices by 20%. In week 2, the "high price" manager raises prices by 30%, and so forth. Suppose an item started out at $100.A store has two managers, one who believes that high profits come from lowering prices and getting more customers and another who
Using the histograms (from Section 6.6, Exercises 9-12), estimate the median, the mode, and the expectation.1.2.3.4.
The p.d.f. of a random variable X is f(x) = 2x for 0 ≤ x ≤ 1. Find the median and mode of the continuous random variables with the given p.d.f., and compare with the expectation as found earlier.
Experiment a. 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.
The p.d.f. of a random variable X is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. Find the quartiles of a random variable with the given p.d.f. Illustrate the areas on a graph of the p.d.f.
The p.d.f. of a random variable T is h(t) = 1/t for 1 ≤ t ≤ e. Find the quartiles of a random variable with the given p.d.f. Illustrate the areas on a graph of the p.d.f.
The p.d.f. of a random variable T is g(t) = 6t(l - t) for 0 ≤ t ≤ 1. This requires a computer (or Newton's method) to solve the equations. Find the quartiles of a random variable with the given p.d.f. Illustrate the areas on a graph of the p.d.f.
The p.d.f. of a random variable X is f(x) = 2x for 0 ≤ x ≤ 1. Find the variance and standard deviation of a continuous random variable with the given p.d.f.
The p.d.f. of a random variable X is f(x) = l - x/2 for 0 ≤ x ≤ 2. Find the variance and standard deviation of a continuous random variable with the given p.d.f.
The p.d.f. of a random variable T is h(t) = 1/t for 1 ≤ t ≤ e. Find the variance and standard deviation of a continuous random variable with the given p.d.f.
The p.d.f. of a random variable T is g(t) = 6t(1 - t) for 0 ≤ t ≤ 1. Find the variance and standard deviation of a continuous random variable with the given p.d.f.
The p.d.f. of a random variable X is f(x) = 2x for 0 ≤ x ≤ l. Find MAD for a continuous random variable with the given p.d.f. How does it compare with the standard deviation found in the earlier problem?
The p.d.f. of a random variable X is f(x) = l - x/2 for 0 ≤ x ≤ 2. Find MAD for a continuous random variable with the given p.d.f. How does it compare with the standard deviation found in the earlier problem?
The p.d.f. of a random variable X is f(x) = 2x for 0 ≤ x ≤ l. Find the probability that the random variable has a value less than one standard deviation below the expectation and less than two standard deviations below the expectation. How do the results compare with the rules of thumb for a
Experiment b. 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.
The p.d.f. of a random variable X is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. Find the probability that the random variable has a value less than one standard deviation below the expectation and less than two standard deviations below the expectation. How do the results compare with the rules of thumb for
The p.d.f. of a random variable T is h(t) = 1/t for 1 ≤ t ≤ e. Find the probability that the random variable has a value less than one standard deviation below the expectation and less than two standard deviations below the expectation. How do the results compare with the rules of thumb for a
The p.d.f. of a random variable T is g(t) = 6t(1 - t) for 0 ≤ t ≤ 1 Find the probability that the random variable has a value less than one standard deviation below the expectation and less than two standard deviations below the expectation. How do the results compare with the rules of thumb
Multiply out the squared term into three terms and break the sum into three sums. The above steps outline the proof of the computational formula for the variance.
To further simplify, try the following steps. a. Factor constants out of the sums. b. Remember that is a constant. c. Recognize certain sums to be equal to the mean. d. Write in terms of the mean and group together like terms. The above steps outline the proof of the computational formula for the

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