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Questions and Answers of Statistics

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
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
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
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
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
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.
A p.d.f. with mean of 50 and coefficient of variation of 0.1. 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
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.
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
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
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
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
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
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
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.
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.
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
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
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 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
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) =
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
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
(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
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
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
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
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
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
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
(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
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
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
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
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
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
(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
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
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
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
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
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
(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
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
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

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