7.3.1 For the following joint distributions, find the probabilities for the random variable X + Y and check that E(X + Y ) = E(X) + E(Y ). (from Section 7.1, Exercises 1 and 5) ...
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7.3.2 For the following joint distributions, find the probabilities for the random variable X + Y and check that E(X + Y ) = E(X) + E(Y ). (from Section 7.1, Exercises 2 and 6) ...
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7.3.3 For the following joint distributions, find the probabilities for the random variable X + Y and check that E(X + Y ) = E(X) + E(Y ). (from Section 7.1, Exercises 3 and 7) ...
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7.3.4 For the following joint distributions, find the probabilities for the random variable X + Y and check that E(X + Y ) = E(X) + E(Y ). (from Section 7.1, Exercises 4 and 8) ...
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7.3.6 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.
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7.3.6 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.
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7.3.7 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.
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7.3.8 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.
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7.3.9 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 1.
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7.3.10 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 2, with variances found in Section 7.2, Exercise 10.
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7.3.11 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 3, with variances found in Section 7.2, Exercise 11.
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7.3.12 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.
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7.3.13 Consider any random variable X that has a finite expectation and variance. Find Cov(X, X). How does it compare with Var(X)?
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7.3.14 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)?
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7.3.15 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 the mean and variance of S directly from its probability distribution.
c. Find E(Z) by using Theorem 7.4 and Var(Z) by using the general addition rule for covariances and the covariance of X with itself (Exercise 13).
d. Find E(Z) by using the fact that Z =2X and Theorem 7.5. and Var(Z) by using the fact that Z =2X and Theorem 7.11.
e. Find E(S) and Var(S) from Theorem 7.4 and Theorem 7.9. Why is Var(S)<Var(Z)? X and Y take the value 0 with probability 0.5 and the value 1 with probability 0.5.
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7.3.16 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 the mean and variance of S directly from its probability distribution.
c. Find E(Z) by using Theorem 7.4 and Var(Z) by using the general addition rule for covariances and the covariance of X with itself (Exercise 13).
d. Find E(Z) by using the fact that Z =2X and Theorem 7.5. and Var(Z) by using the fact that Z =2X and Theorem 7.11.
e. Find E(S) and Var(S) from Theorem 7.4 and Theorem 7.9. Why is Var(S)<Var(Z)? 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.
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7.3.19 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. Use Theorem 7.4 to break up the expectations. Theorem 7.4 ...
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7.3.19 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. Use Theorem 7.4 to break up the expectations. Theorem 7.4 ...
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7.3.19 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. Use Theorem 7.4 to break up the expectations. Theorem 7.4 ...
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7.3.20 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. Use a law of exponents
and the definition of the geometric mean to prove the result.
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7.3.21 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. X takes the value 1 with probability 0.5 and the value 2 with probability 0.5.
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7.3.22 The harmonic mean is another kind of average (like the geometric mean), defined by ... (the
reciprocal of the expectation of the reciprocal). Like the geometric
mean, the harmonic mean is only defined for random variables that take
on positive values, and it is always less than the expectation. Compute
the harmonic mean of the following random variables and check that it is
indeed less than the expectation. Suppose X takes the value 1 with probability 0.1 and the value 10 with probability 0.9.
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7.3.23 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 two random variables are independent. Use the
harmonic mean (Exercises 21 and 22) to guess why the expectation of the
quotient is bigger than the quotient of the expectations. 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.
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7.3.24 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 two random variables are independent. Use the
harmonic mean (Exercises 21 and 22) to guess why the expectation of the
quotient is bigger than the quotient of the expectations. 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
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7.3.26 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.
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7.3.27 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 C.
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7.3.27 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 C.
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7.3.28 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.
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7.3.30 Find the probability distribution of W =0.05L + 0.02M for the following birds. Check that the expectation is 0.059 in both cases. Bird B (from Table 7.1). Table 7.1 ...
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7.3.30 Find the probability distribution of W =0.05L + 0.02M for the following birds. Check that the expectation is 0.059 in both cases. Bird B (from Table 7.1). Table 7.1 ...
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7.3.31
Suppose
immigration and emigration change the sizes of four populations with
the following probabilities (Section 6.7, Exercises 31–34). ...
Find the expectation of the population change in populations a and c
summed.
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7.3.32
Suppose
immigration and emigration change the sizes of four populations with
the following probabilities (Section 6.7, Exercises 31–34). ...
Find the expectation of the total population change in b and d summed.
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7.3.33
Suppose annual immigration into a park by three species follows the
probabilities in the table. ... Find the expected number of each
type of immigrant. Find the expected total number of immigrants from
all species combined.
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7.3.34 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.
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7.3.35
Suppose annual immigration into a park by three species follows the
probabilities in the table. ... 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 species and compare with the sum of the expected numbers.
d. Find the variance in the total number of immigrants and compare with the sum of the variances.
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7.3.37 Suppose annual immigration into a park by three species follows the probabilities in the table. ... 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.
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7.3.37 Suppose annual immigration into a park by three species follows the probabilities in the table. ... 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.
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7.3.38 Suppose annual immigration into a park by three species follows the probabilities in the table. ... 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.
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7.3.39
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/...) of 1.0, and the second
has food quality of 2.0. The two volumes are 0.5 and 1.5 ... ...
Which of the joint distributions is independent?
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7.3.40
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/...) of 1.0, and the second
has food quality of 2.0. The two volumes are 0.5 and 1.5 ... ...
Find the mean quality and volume of caterpillars eaten by each bird.
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7.3.42
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/...) of 1.0, and the second
has food quality of 2.0. The two volumes are 0.5 and 1.5 ... ...
From the expected total calories for each bird, and find the covariance
of quality and volume.
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7.3.42
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/...) of 1.0, and the second
has food quality of 2.0. The two volumes are 0.5 and 1.5 ... ...
From the expected total calories for each bird, and find the covariance
of quality and volume.
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7.3.43 Consider the following data for cell age A and the number of toxic molecules N inside (as in Section 7.2, Exercises 37 and 38). ... 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
find the overall probability that a cell is cancerous, and show how it
is really a version of the sum rule for expectations. Suppose that the probability that a cell is cancerous is ...
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7.3.44 Consider the following data for cell age A and the number of toxic molecules N inside (as in Section 7.2, Exercises 37 and 38). ... 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
find the overall probability that a cell is cancerous, and show how it
is really a version of the sum rule for expectations.` Suppose that the probability that a cell is cancerous is ...
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7.3.45 Consider Markov chains where ... represents the event that a molecule is outside a cell at time t, obeying ... for various values of p.
In each case, the long-term probability that the molecule is inside is
0.5. Suppose you get one point when the molecule is inside and zero
points when it is outside.
a. ...
b. What should the value be? Which value is closest?
c. Try again with the initial value ...set to 1 with probability 0.5 and to 0 with probability 0.5. Does this change your results?
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7.3.46 Recall
the dice rolling rules in Section 7.1, Exercise 41. Redo the
experiment, and find the average value for each die. Is the sum of the
averages equal to the average of the sums?
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