8.1.1 Suppose
we wish to calculate the proportion of days that the temperature rises
above 20?C. Evaluate the following sampling schemes. Sample 100 consecutive days beginning on January 1.
Get solution
8.1.2 Suppose
we wish to calculate the proportion of days that the temperature rises
above 20?C. Evaluate the following sampling schemes. Sample 100 consecutive days beginning on June 1.
Get solution
8.1.3 Suppose
we wish to calculate the proportion of days that the temperature rises
above 20?C. Evaluate the following sampling schemes. Sample the temperature on March 15 for 100 years.
Get solution
8.1.4 Suppose
we wish to calculate the proportion of days that the temperature rises
above 20?C. Evaluate the following sampling schemes. What might be a good method if only 100 days could be sampled?
Get solution
8.1.5 A
clever pollster decides to find the average income of people by calling
random individuals on the phone. It turns out, however, that people
with cellular phones make $40,000 and people with regular phones make
$20,000. Furthermore, 20% of people have cellular phones. What is the true average income in this population?
Get solution
8.1.6
A
clever pollster decides to find the average income of people by calling
random individuals on the phone. It turns out, however, that people
with cellular phones make $40,000 and people with regular phones make
$20,000. Furthermore, 20% of people have cellular phones. What income
would be estimated if people with cell phones were easier to reach, and
50% of the people called had cell phones?
Get solution
8.1.7 Consider
again the situation in Exercises 5 and 6, but suppose that incomes for
people with cell phones are normally distributed according to N (40000, 4.0 ×...) (measured in 1999 U.S. dollars). The distribution for people without cellular phones is N(20000, 1.0 ×...). What
distribution describes the result of sampling 20 people with cellular
phones? Use the rule of thumb that 95% of the distribution lies within
two standard deviations of the mean to give a probable range. Compare
this with the true average of the population.
Get solution
8.1.8 Consider
again the situation in Exercises 5 and 6, but suppose that incomes for
people with cell phones are normally distributed according to N (40000, 4.0 ×...) (measured in 1999 U.S. dollars). The distribution for people without cellular phones is N(20000, 1.0 ×...). What
would be the results of sampling 20 people without cellular phones? Use
the rule of thumb that 95% of the distribution lies within two standard
deviations of the mean to give a probable range. Compare this with the
true average of the population.
Get solution
8.1.9 Find the likelihood as a function of the binomial proportion p for each of the following. Flipping 2 out of 4 heads with a fair coin. Evaluate at p = 0.5, the value for a fair coin.
Get solution
8.1.10 Find the likelihood as a function of the binomial proportion p for each of the following. Rolling 2 out of 4 sixes with a fair die. Evaluate at p = 1/6, the value for a fair die.
Get solution
8.1.11 Find the likelihood as a function of the binomial proportion p for each of the following. Flipping 2 out of 12 heads with a fair coin.
Get solution
8.1.12 Find the likelihood as a function of the binomial proportion p for each of the following. Rolling 2 out of 12 sixes with a fair die.
Get solution
8.1.13 Find
the probability distribution associated with the following random
variables, and identify which part corresponds to the likelihood found
in the earlier problem. Let H represent the number of heads in four flips of a fair coin. Compare with Exercise 9.
Get solution
8.1.14 Find
the probability distribution associated with the following random
variables, and identify which part corresponds to the likelihood found
in the earlier problem. The number N of sixes rolled in four rolls of a fair die. Compare with Exercise 10.
Get solution
8.1.15 Find the likelihood as a function of the binomial proportion p in each of the following cases, and find the maximum likelihood. Team
A wins five out of six games in a series against team B. Find the
maximum likelihood estimator of the probability that team A wins a game
against team B. If you were willing to gamble, would it make sense to
enter a bet about the next game in the series where you win $1 if team A
wins, but lose $6 if team A loses?
Get solution
8.1.16 Find the likelihood as a function of the binomial proportion p in each of the following cases, and find the maximum likelihood. One
out of 150 people you know wins $500 in a raffle that costs $5 to
enter. Find the maximum likelihood estimator of the probability of
winning the raffle. What is your best guess of the average payoff?
Get solution
8.1.17
Find
the likelihood as a function of the Poisson parameter Λ, find the
maximum likelihood estimator, and evaluate the likelihood at the maximum
and at the other given value of Λ. Twenty events occur in 1 min.
Compare the likelihood with the maximum likelihood estimator of Λ with
the likelihood if Λ= 10.0.
Get solution
8.1.18 Find
the likelihood as a function of the Poisson parameter Λ, find the
maximum likelihood estimator, and evaluate the likelihood at the maximum
and at the other given value of Λ. Ten high energy cosmic rays
hit detector over the course of 1 yr. Compare the likelihood with the
maximum likelihood estimator of Λ with the likelihood if Λ = 8.0.
Get solution
8.1.19 Find
the likelihood as a function of the Poisson parameter Λ, find the
maximum likelihood estimator, and evaluate the likelihood at the maximum
and at the other given value of Λ. The number of events that
occur are counted for 3 min. Twenty events occur the first minute, 16
events occur the second minute, and 21 events occur the third minute.
Compare the likelihood with the maximum likelihood estimator of Λ with the likelihood if Λ= 20.0.
Get solution
8.1.20 Find
the likelihood as a function of the Poisson parameter Λ, find the
maximum likelihood estimator, and evaluate the likelihood at the maximum
and at the other given value of Λ. Ten high energy cosmic rays
hit detector in its first year, 7 in the second year, 11 in the third
year, and 8 in the fourth and final year. Compare the likelihood with
the maximum likelihood estimator of Λ with the likelihood if Λ = 10.0.
Get solution
8.1.21 Find the likelihood as a function of the parameter q of
a geometric distribution, find the maximum likelihood estimator, and
evaluate the likelihood at the maximum and at the other given value of q. Flies
are tested for the ability to learn to fly toward the smell of potato,
and the first to succeed is the 13th. Compare with the likelihood of q = 0.1.
Get solution
8.1.22 Find the likelihood as a function of the parameter q of
a geometric distribution, find the maximum likelihood estimator, and
evaluate the likelihood at the maximum and at the other given value of q. Random
compounds are tested for the ability to suppress a particular type of
tumor, and the first to succeed is the 94th. Compare with the likelihood
of q = 0.005.
Get solution
8.1.23 Find the likelihood as a function of the parameter q of
a geometric distribution, find the maximum likelihood estimator, and
evaluate the likelihood at the maximum and at the other given value of q. The
experiment testing flies for the ability to learn to fly toward the
smell of potato is repeated three times. In the first experiment the
first fly to succeed is the 13th, in the second experiment it is the
8th, and in the third experiment it is the 12th. Compare with the
likelihood of q = 0.1.
Get solution
8.1.24 Find the likelihood as a function of the parameter q of
a geometric distribution, find the maximum likelihood estimator, and
evaluate the likelihood at the maximum and at the other given value of q. The
experiment testing compounds for the ability to suppress tumors is
repeated twice. In the first experiment the first compound to succeed
is the 94th, and in the second experiment it is the 406th. Compare with
the likelihood of q = 0.005.
Get solution
8.1.25 Write down the equations that would express the fact that the following estimators are unbiased. The estimator of q in Exercise 21.
Get solution
8.1.26 Write down the equations that would express the fact that the following estimators are unbiased. The estimator of Λ in
Exercise 17. If you think about the definition of the expectation, you
might be able to demonstrate that this estimator is unbiased.
Get solution
8.1.27 In
each of the following cases where a very small number of individuals is
tested for an allele, find and graph the likelihood function for the
proportion p of individuals in the whole population with this
allele, find the maximum likelihood estimator, and make sense of the
likelihood at p = 0 and p = 1. Tw o individuals are tested for a particular allele and one has it.
Get solution
8.1.28 In
each of the following cases where a very small number of individuals is
tested for an allele, find and graph the likelihood function for the
proportion p of individuals in the whole population with this
allele, find the maximum likelihood estimator, and make sense of the
likelihood at p = 0 and p = 1. Three individuals are tested for a particular allele, and all three have it.
Get solution
8.1.29
Thirty
out of 100 individuals are found to be infected with a disease.
Estimate the proportion of infected women and infected men in the
following circumstances. Assuming that the whole population is composed
of 50% women, estimate the infected proportion in the whole population.
The sample consists of 20 out of 50 infected women and 10 out of 50
infected men.
Get solution
8.1.30
Thirty
out of 100 individuals are found to be infected with a disease.
Estimate the proportion of infected women and infected men in the
following circumstances. Assuming that the whole population is composed
of 50% women, estimate the infected proportion in the whole population.
The sample consists of 20 out of 40 infected women and 10 out of 60
infected men.
Get solution
8.1.31
Thirty
out of 100 individuals are found to be infected with a disease.
Estimate the proportion of infected women and infected men in the
following circumstances. Assuming that the whole population is composed
of 50% women, estimate the infected proportion in the whole population.
The sample consists of 20 out of 20 infected women and 10 out of 80
infected men.
Get solution
8.1.32
Thirty
out of 100 individuals are found to be infected with a disease.
Estimate the proportion of infected women and infected men in the
following circumstances. Assuming that the whole population is composed
of 50% women, estimate the infected proportion in the whole population.
The sample consists of 0 out of 50 infected women and 30 out of 50
infected men.
Get solution
8.1.33 Two couples are trying to have more girl babies. For each, find the likelihood function for the fraction q of female sperm and the maximum likelihood estimator, and compare with the likelihood of q = 0.5.
The first couple has seven boys before having a girl. Use the
geometric distribution to build the likelihood as a function of q.
Get solution
8.1.34 Two couples are trying to have more girl babies. For each, find the likelihood function for the fraction q of female sperm and the maximum likelihood estimator, and compare with the likelihood of q = 0.5. Another couple has four boys, then one girl, then three more boys. Find the likelihood as a function of q.
Get solution
8.1.35 Use the method of maximum likelihood to estimate the rate λ from the accompanying table of data drawn from an exponential distribution. ... In
each case, find the likelihood function, find the maximum likelihood,
and say whether it seems probable that the true rate is 1.0. For waiting time 1.
Get solution
8.1.36 Use the method of maximum likelihood to estimate the rate λ from the accompanying table of data drawn from an exponential distribution. ... In
each case, find the likelihood function, find the maximum likelihood,
and say whether it seems probable that the true rate is 1.0. For waiting time 2.
Get solution
8.1.37 Mutations
are counted in four pieces of DNA that are 1 million base pairs long.
There are 14 mutations in the first piece, 17 in the second piece, 8 in
the third piece, and 5 in the fourth piece. Write the likelihood
function for the expected number of mutations per million bases in the
first piece and find the maximum likelihood estimator.
Get solution
8.1.38 Mutations
are counted in four pieces of DNA that are 1 million base pairs long.
There are 14 mutations in the first piece, 17 in the second piece, 8 in
the third piece, and 5 in the fourth piece. Write the likelihood
function for the expected number of mutations per million bases in the
second piece and find the maximum likelihood estimator.
Get solution
8.1.39 Mutations
are counted in four pieces of DNA that are 1 million base pairs long.
There are 14 mutations in the first piece, 17 in the second piece, 8 in
the third piece, and 5 in the fourth piece. Write the likelihood
function for the expected number of mutations per million bases in the
first two pieces and find the maximum likelihood estimator. Compare this
with the estimated expected number for each of the two pieces
separately.
Get solution
8.1.40 Mutations
are counted in four pieces of DNA that are 1 million base pairs long.
There are 14 mutations in the first piece, 17 in the second piece, 8 in
the third piece, and 5 in the fourth piece. Write the likelihood
function for the expected number of mutations per million bases in the
first four pieces and find the maximum likelihood estimator. Compare
this with the estimated expected number for each of the four pieces
separately.
Get solution
8.1.41 Mutation
rates differ depending on whether changing the nucleotide base changes
the amino acid (nonsynonymous sites) or not (synonymous sites). A piece
of DNA has 200 nonsynonymous sites with 12 mutations and 100 synonymous
sites with 15 mutations. Our goal is to estimate the mutation rate. Estimate ... , the mutation rate for synonymous sites, from the given data.
Get solution
8.1.42 Mutation
rates differ depending on whether changing the nucleotide base changes
the amino acid (nonsynonymous sites) or not (synonymous sites). A piece
of DNA has 200 nonsynonymous sites with 12 mutations and 100 synonymous
sites with 15 mutations. Our goal is to estimate the mutation rate. Estimate ...the mutation rate for nonsynonymous sites, from the given data.
Get solution
8.1.43
Mutation
rates differ depending on whether changing the nucleotide base changes
the amino acid (nonsynonymous sites) or not (synonymous sites). A piece
of DNA has 200 nonsynonymous sites with 12 mutations and 100 synonymous
sites with 15 mutations. Our goal is to estimate the mutation rate.
Suppose we assume that the synonymous and nonsynonymous rates are both
equal to the same value λ. Estimate λ, and compare with the values of ...found in Exercises 41 and 42.
Get solution
8.1.44
Mutation
rates differ depending on whether changing the nucleotide base changes
the amino acid (nonsynonymous sites) or not (synonymous sites). A piece
of DNA has 200 nonsynonymous sites with 12 mutations and 100 synonymous
sites with 15 mutations. Our goal is to estimate the mutation rate.
Suppose instead that the synonymous rate is three times that of the
nonsynonymous rate. Formally, ...Estimate λ, and compare with the values of λ found in Exercises 41 and 42.
Get solution
8.1.45 Color blindness is due to a recessive allele that appears on the X chromosome. If the color blindness allele has frequency p, a fraction p of males will show the phenotype, and a fraction ... of females will show the phenotype (because they require two copies). We wish to estimate the fraction p from a sample of 1000 males, 90 of whom are color-blind, and 1000 females, 13 of whom are color-blind. Estimate p using just the males.
Get solution
8.1.46 Color blindness is due to a recessive allele that appears on the X chromosome. If the color blindness allele has frequency p, a fraction p of males will show the phenotype, and a fraction ... of females will show the phenotype (because they require two copies). We wish to estimate the fraction p from a sample of 1000 males, 90 of whom are color-blind, and 1000 females, 13 of whom are color-blind. Estimate p using just the females.
Get solution
8.1.47 Color blindness is due to a recessive allele that appears on the X chromosome. If the color blindness allele has frequency p, a fraction p of males will show the phenotype, and a fraction ... of females will show the phenotype (because they require two copies). We wish to estimate the fraction p from
a sample of 1000 males, 90 of whom are color-blind, and 1000 females,
13 of whom are color-blind. Write the likelihood function for males and
females together.
Get solution
8.1.48 Color blindness is due to a recessive allele that appears on the X chromosome. If the color blindness allele has frequency p, a fraction p of males will show the phenotype, and a fraction ... of females will show the phenotype (because they require two copies). We wish to estimate the fraction p from
a sample of 1000 males, 90 of whom are color-blind, and 1000 females,
13 of whom are color-blind. Evaluate the likelihood function in
Exercise 47 at p = 0.09, p = 0.114, and p = 0.1. Where do you think the maximum might be? If you are very determined, it is possible to solve for the maximum.
Get solution
8.1.49 Simulate the following experiments. a.
People have a 0.4 chance of having black hair. Simulate 50 groups of
five people. How many groups have exactly two out of five people with
black hair?
b. Suppose instead that two out of five people are
found with black hair. Simulate 50 groups of five people with different
unknown probabilities p that a person has black hair. Use values of p ranging from 0 to 1.0. Which value produces the most groups that exactly match the data? c.
Find and plot the likelihood function describing this case. Where is
the maximum? What does it correspond to in your simulation?
d. Explain how the two simulations differ. In each case, indicate what is known and what is unknown.
Get solution
8.1.50 Cells
are placed for 1 min in an environment where they are hit by X-rays,
some of which are damaging. Cells not hit by the damaging rays are
healthy, those hit exactly once are damaged, and those hit more than
once are dead. By measuring the states of a number of cells, we wish to
infer the rate at which cells are hit by damaging rays. Let x denote
the unknown parameter of the Poisson distribution. Use the formula for
the Poisson distribution to compute the probabilities ...of more than one hit in 1 min as functions of x.
a. Suppose the true value of x is 3.0. Plot the histogram. b.
Simulate 50 cells, and count how many you have of each type. (To keep
things interesting, keep sampling until you get at least one cell of
each type.)
c. Compare the results of your simulation with the idealized histogram.
d. Now pretend that x is unknown. We can use the method of maximum likelihood to analyze our data. Find the likelihood function L of these data (it is the product of the likelihoods for each of the 50 cells) as a function of the unknown parameter y, and let S be the natural log of L. Plot S(y) over a reasonable range.
e. Find the maximum of S and mark it on your graph.
f. Find the S(x) for x = ..., x = 2, x = 4, and the “truth” x = 3, and indicate each on your graph.
Get solution
