7.2.1 For the following joint distributions, find the covariance of X and Y using the direct method, Cov(X, Y )= ... If the covariance is zero, are the random variables independent? (from Section 7.1, Exercises 1 and 5) ...
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7.2.2 For the following joint distributions, find the covariance of X and Y using the direct method, Cov(X, Y )= ... If the covariance is zero, are the random variables independent? (from Section 7.1, Exercises 2 and 6) ...
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7.2.3 For the following joint distributions, find the covariance of X and Y using the direct method, Cov(X, Y )= ... If the covariance is zero, are the random variables independent? (from Section 7.1, Exercises 3 and 7) ...
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7.2.4 For the following joint distributions, find the covariance of X and Y using the direct method, Cov(X, Y )= ... If the covariance is zero, are the random variables independent? (from Section 7.1, Exercises 4 and 8) ...
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7.2.5 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 1.
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7.2.6 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.
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7.2.7 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.
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7.2.9 For the following joint distributions, find the correlation of X and Y . The case in Exercise 1.
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7.2.10 For the following joint distributions, find the correlation of X and Y . The case in Exercise 2.
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7.2.10 For the following joint distributions, find the correlation of X and Y . The case in Exercise 2.
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7.2.11 For the following joint distributions, find the correlation of X and Y . The case in Exercise 3.
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7.2.12 For the following joint distributions, find the correlation of X and Y . The case in Exercise 4.
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7.2.13 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 1
point and that a miss is worth 0. (Let ... be a random variable for the first player and ... be a random variable for the second player; let ...
=1 mean that player 1 got a hit, and so forth.) Say why the covariance
is positive or negative. The case where the players have the highest
possible probability of both getting a hit (as in Section 7.1, Exercise
19).
Get solution
7.2.14 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 1
point and that a miss is worth 0. (Let ... be a random variable for the first player and ... be a random variable for the second player; let ...
=1 mean that player 1 got a hit, and so forth.) Say why the covariance
is positive or negative. The case where the players have the highest
possible probability of both getting a hit (as in Section 7.1, Exercise
19). The case where the players have the lowest possible probability of
both getting a hit (as in Section 7.1, Exercise 20).
Get solution
7.2.15 In
most Markov chains, the state of the system is correlated from step to
step. Using the joint distribution of the states of the following Markov
chains at times t and t + 1, find the correlation (set
the value of one state to be 1 and of the other state to be 0). Assume
that the marginal distributions at both time t and t + 1 match the long-term probability. 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.
Get solution
7.2.16 In
most Markov chains, the state of the system is correlated from step to
step. Using the joint distribution of the states of the following Markov
chains at times t and t + 1, find the correlation (set
the value of one state to be 1 and of the other state to be 0). Assume
that the marginal distributions at both time t and t + 1 match the long-term probability. 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.
Get solution
7.2.18 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, between X and Y that includes the range of values X =0, X =1, and X =2. ...
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7.2.18 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, between X and Y that includes the range of values X =0, X =1, and X =2. ...
Get solution
7.2.19 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, between X and Y that includes the range of values X =0, X =1, and X =2. ...
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7.2.21
Find the covariance of the number of lice and mites for the following
birds from the earlier problem. Bird C from Section 7.1, Exercise 23.
Get solution
7.2.21
Find the covariance of the number of lice and mites for the following
birds from the earlier problem. Bird C from Section 7.1, Exercise 23.
Get solution
7.2.22
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.
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7.2.23 Find the correlation of the number of lice and mites for the following birds. Bird C from Exercise 21.
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7.2.24 Find the correlation of the number of lice and mites for the following birds. Bird D from Exercise 22.
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7.2.25 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 no eagle, E =1 seeing an eagle, J =0 seeing no jackrabbit, and J =1 seeing a jackrabbit. What does the result mean? 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 (as in
Section 7.1, Exercise 27).
Get solution
7.2.26 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 no eagle, E =1 seeing an eagle, J =0 seeing no jackrabbit, and J =1 seeing a jackrabbit. What does the result mean? 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 (as in
Section 7.1, Exercise 28).
Get solution
7.2.27 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 properly with probability 0.5. Suppose Pr(cell is 0 days 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. Using the random variables A for age and S for
staining, find the covariance in the following cases. Explain why the
covariance is positive or negative. With the probabilities as given (as
in Section 7.1, Exercise 33).
Get solution
7.2.28 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 properly with probability 0.5. Suppose Pr(cell is 0 days 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. Using the random variables A for age and S for
staining, find the covariance in the following cases. Explain why the
covariance is positive or negative. If the lab finds a way to eliminate
the oldest cells (>3 days old) from its stock (as in Exercise 34).
Get solution
7.2.29 Consider birds that spend all of their time sleeping, eating, or preening. Let S be the time spent sleeping, E the time spent eating, and P be
the time spent preening. How much time is spent on each activity
depends on the weather, which is nice with probability 0.7, OK with
probability 0.2, and terrible with probability 0.1. 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 ρS,P .
Get solution
7.2.30 Consider birds that spend all of their time sleeping, eating, or preening. Let S be the time spent sleeping, E the time spent eating, and P be
the time spent preening. How much time is spent on each activity
depends on the weather, which is nice with probability 0.7, OK with
probability 0.2, and terrible with probability 0.1. 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 ρS,P . Why are ρE,P and ρE,S not worth finding?
Get solution
7.2.31 Consider birds that spend all of their time sleeping, eating, or preening. Let S be the time spent sleeping, E the time spent eating, and P be
the time spent preening. How much time is spent on each activity
depends on the weather, which is nice with probability 0.7, OK with
probability 0.2, and terrible with probability 0.1. 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 ρS,P . Why is the correlation not perfect?
Get solution
7.2.32 Consider birds that spend all of their time sleeping, eating, or preening. Let S be the time spent sleeping, E the time spent eating, and P be
the time spent preening. How much time is spent on each activity
depends on the weather, which is nice with probability 0.7, OK with
probability 0.2, and terrible with probability 0.1. 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 ρS,P . Why are ρE,P and ρE,S not worth finding? 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 ρS,P . Why is the correlation not perfect?
Get solution
7.2.33
Suppose
immigration and emigration change the sizes of two populations with the
following probabilities (as in Section 6.7, Exercises 31 and 32).
... Let ...represent the change in population a and ...the change in
population
b. Explicitly compute the covariance if immigrants enter the two populations independently (as in Section 7.1, Exercise 35).
Get solution
7.2.34
Suppose
immigration and emigration change the sizes of two populations with the
following probabilities (as in Section 6.7, Exercises 31 and 32).
... Let ...represent the change in population a and ...the change in
population
b. Compute the covariance in the case from Section 7.1, Exercise 36.
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7.2.35 Suppose the following are measurements of the temperature T and insect size S. ... Find the correlation of S with T .
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7.2.37 Consider the following data for cell age A and the number of toxic molecules N inside. ... Find the correlation of A with N.
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7.2.37 Consider the following data for cell age A and the number of toxic molecules N inside. ... Find the correlation of A with N.
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7.2.38 Consider the following data for cell age A and the number of toxic molecules N inside. ... Suppose that the damage done by the toxic molecules is D = ln(1 + N). Find the correlation of A with D.
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