7.1.1 For the following 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? Find the marginal
distributions and the distribution of Y conditional on X =0. ...
Get solution
7.1.2 For the following 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? Find the marginal
distributions and the distribution of X conditional on Y =1. ...
Get solution
7.1.3 For the following 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? Find the marginal
distributions and the distribution of X conditional on Y =1. ...
Get solution
7.1.4 For the following 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? Find the marginal
distributions and the distribution of Y conditional on X =2. ...
Get solution
7.1.6 Find the expectations of the random variables from their marginal distributions. The random variables X and Y in Exercise 2.
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7.1.6 Find the expectations of the random variables from their marginal distributions. The random variables X and Y in Exercise 2.
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7.1.7 Find the expectations of the random variables from their marginal distributions. The random variables X and Y in Exercise 3.
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7.1.8 Find the expectations of the random variables from their marginal distributions. The random variables X and Y in Exercise 4.
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7.1.9 Suppose
the following random variables are independent. Find the joint
distribution and the requested conditional distribution from the given
marginal distributions. 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.
Get solution
7.1.11 Suppose
the following random variables are independent. Find the joint
distribution and the requested conditional distribution from the given
marginal distributions. 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.
Get solution
7.1.11 Suppose
the following random variables are independent. Find the joint
distribution and the requested conditional distribution from the given
marginal distributions. 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.
Get solution
7.1.12 Suppose
the following random variables are independent. Find the joint
distribution and the requested conditional distribution from the given
marginal distributions. The random variable X has probability distribution Pr(X = 0)=0.3, Pr(X =1)=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.
Get solution
7.1.13
Use the given information to construct the entire joint distribution
for the following pairs of random variables. 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.
Get solution
7.1.14
Use the given information to construct the entire joint distribution
for the following 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 =1)=0.5, and Pr(X =1 and Y =0)=0.4.
Get solution
7.1.15
Use the given information to construct the entire joint distribution
for the following 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.
Get solution
7.1.16
Use the given information to construct the entire joint distribution
for the following 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.
Get solution
7.1.17 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 =0 and N >0. Find the joint distribution for these events.
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7.1.19 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
distribution for the second player conditional on the first getting a
hit, and the conditional distribution for the first player conditional
on the second getting a hit. The case where the players have the highest possible probability of each getting a hit.
Get solution
7.1.19 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
distribution for the second player conditional on the first getting a
hit, and the conditional distribution for the first player conditional
on the second getting a hit. The case where the players have the highest possible probability of each getting a hit.
Get solution
7.1.20 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
distribution for the second player conditional on the first getting a
hit, and the conditional distribution for the first player conditional
on the second getting a hit. The case where the players have the lowest possible probability of each getting a hit
Get solution
7.1.22 Write the joint distribution describing the states of the following Markov chains at times t and t + 1. Assume that the marginal distributions at both time t and t + 1 match the longterm probability. 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.
Get solution
7.1.22 Write the joint distribution describing the states of the following Markov chains at times t and t + 1. Assume that the marginal distributions at both time t and t + 1 match the longterm probability. 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.
Get solution
7.1.24 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
Get solution
7.1.24 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
Get solution
7.1.25 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 C, from Exercise 23.
Get solution
7.1.26 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.
Get solution
7.1.27 Recall
the ecologist observing eagles and rabbits (Section 6.4, Exercises 27
and 28). In each of the following cases, find the joint distribution,
the marginal distributions, and the conditional distributions. Use 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. Use graphs of the conditional distributions to help interpret the results. 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 (as in Section 6.4, Exercise 27).
Get solution
7.1.28 Recall
the ecologist observing eagles and rabbits (Section 6.4, Exercises 27
and 28). In each of the following cases, find the joint distribution,
the marginal distributions, and the conditional distributions. Use 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. Use graphs of the conditional distributions to help interpret the results. 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 (as in Section 6.4, Exercise 28).
Get solution
7.1.30 Find
the joint distribution of the two events in the rare disease model
(Section 6.4, page 523) where a person either has the disease (event D)
or does not (event N) and either tests positive (event P) or does not
(event ...) in the following cases. Use the joint distribution to find Pr(D | P). Pr(D)=0.8, Pr(P |D)=1.0, and Pr(P |N)=0.1 (as in Section 6.4, Exercise 38).
Get solution
7.1.30 Find
the joint distribution of the two events in the rare disease model
(Section 6.4, page 523) where a person either has the disease (event D)
or does not (event N) and either tests positive (event P) or does not
(event ...) in the following cases. Use the joint distribution to find Pr(D | P). Pr(D)=0.8, Pr(P |D)=1.0, and Pr(P |N)=0.1 (as in Section 6.4, Exercise 38).
Get solution
7.1.31 Find
the joint distribution of the two events in the rare disease model
(Section 6.4, page 523) where a person either has the disease (event D)
or does not (event N) and either tests positive (event P) or does not
(event ...) in the following cases. Use the joint distribution to find Pr(D | P). Pr(D)=0.2, Pr(N)=0.8, Pr(P |D)=0.95, and Pr(P |N)= 0.05 (as in Section 6.4, Exercise 39).
Get solution
7.1.32 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. In
each of the following cases, define two random variables, draw the
joint distribution, find the marginal probability distributions, and
compute and graph the conditional probability distributions for cell
age. Compare the conditional distributions with the marginal
distribution.
Get solution
7.1.33 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 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. Compare the conditional distributions with the marginal distribution with the probabilities as given.
Get solution
7.1.35 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. Find the joint distribution if immigrants enter the two populations independently.
Get solution
7.1.35 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. Find the joint distribution if immigrants enter the two populations independently.
Get solution
7.1.36 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. Fill in the rest of the joint distribution. ...
Get solution
7.1.37
We
have seen that meiotic drive and lack of independence can create
unusual distributions of genotypes for offspring (Section 6.5, Exercises
29 and 30). In the following cases, find the joint distribution of
genotypes in the offspring. Use your joint distribution to find the
probability that an offspring is a heterozygote. 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 0.4 when the ovule provides an
a. The ovule provides A with probability 0.5 (from Section 6.5, Exercise 29).
Get solution
7.1.38
We
have seen that meiotic drive and lack of independence can create
unusual distributions of genotypes for offspring (Section 6.5, Exercises
29 and 30). In the following cases, find the joint distribution of
genotypes in the offspring. Use your joint distribution to find the
probability that an offspring is a heterozygote. 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
probability 0.3 when the ovule provides an
a. The ovule provides A with probability 0.5 (from Section 6.5, Exercise 30).
Get solution
7.1.40 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 going on with these birds? ...
Get solution
7.1.40 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 going on with these birds? ...
Get solution
7.1.41 Suppose
two die are rolled, but the second die must be rerolled until the score
is less than or equal to that on the first. For example, if the first
die rolls a 3, then the second must be rolled again and again until its
value is 3 or less.
a. Roll 100 computer die with these rules and record your results.
b. Try to figure out the mathematical joint distribution for this process.
Get solution
