2.7
Chapter 2 Problems
1. In modelling the number of transactions of a certain type received by a central com- puter for a company with many on-line terminals the Poisson distribution can be used. If the transactions arrive at random at the rate of per minute then the probability of y transactions in a time interval of length t minutes is
P (Y = y; ) = f (y; ) = ( t)
y
y! e
t for y = 0; 1; : : : and > 0:
(a) The numbers of transactions received in 10 separate one minute intervals were 8, 3, 2, 4, 5, 3, 6, 5, 4, 1. Write down the likelihood function for and …nd the maximum likelihood estimate ^.
(b) Estimate the probability that during a two-minute interval, no transactions ar- rive.
(c) Use the R function rpois() with the value = 4:1 to simulate the number of transactions received in 100 one minute intervals. Calculate the sample mean and variance; are they approximately the same? (Note that E(Y ) = V ar(Y ) = for the Poisson model.)
2. Suppose y1; y2; :::yn is an observed random sample from the distribution with proba-
bility density function
f (y) = ( + 1)y for 0 < y < 1 and > 1:
(a) Find the likelihood function, L( ).
(b) Obtain the maximum likelihood estimate of .
3. Consider the following two experiments whose purpose was to estimate , the fraction of a large population with blood type B.
Experiment 1: Individuals were selected at random until 10 with blood type B were found. The total number of people examined was 100.
Experiment 2: One hundred individuals were selected at random and it was found that 10 of them have blood type B.
(a) Find the probability of the observed results (as a function of ) for the two experiments. Thus obtain the likelihood function for for each experiment and show that they are proportional. Show the maximum likelihood estimate ^ is the same in each case. What is the maximum likelihood estimate of ?
(b) Suppose n people came to a blood donor clinic. Assuming = 0:10, how large should n be to ensure that the probability of getting 10 or more B- type donors is at least 0:90? (The R functions gbinom() or pbinom() can help here.)
76 2. STATISTICAL MODELS AND MAXIMUM LIKELIHOOD ESTIMATION
4. Consider Example 2.4.2 on M-N blood types. If a random sample of n individuals gives y1; y2; and y3 persons of types MM, MN, and NN respectively, …nd the maxi-
mum likelihood estimate ^ in the model in terms of y1; y2; y3.
5. Suppose that in a population of twins, males (M ) and females (F ) are equally likely to occur and that the probability that a pair of twins is identical is . If twins are not identical, their sexes are independent.
(a) Show that
P (M M ) = P (F F ) = 1 +
4 and P (M F ) = 1
2
(b) Suppose that n pairs of twins are randomly selected; it is found that n1 are M M ,
n2 are F F , and n3 are M F , but it is not known whether each set is identical or
fraternal. Use these data to …nd the maximum likelihood estimate ^ of . What is the value of ^ if n = 50 and n1 = 16, n2 = 16, n3 = 18?
6. Estimation from capture-recapture studies: In order to estimate the number of animals, N , in a wild habitat the capture-recapture method is often used. In this scheme k animals are caught, tagged, and then released. Later on n animals are caught and the number Y of these that have tags are noted. The idea is to use this information to estimate N .
(a) Show that under suitable assumptions
P (Y = y) = k y N k n y N n
(b) For observed k, n and y …nd the value ^N that maximizes the probability in part (a). Does this ever di¤er much from the intuitive estimate ~N = kn=y? (Hint: The likelihood L(N ) depends on the discrete parameter N , and a good way to …nd where L(N ) is maximized over f1; 2; 3; : : :g is to examine the ratios L(N + 1)=L(N ):)
(c) When might the model in part (a) be unsatisfactory?
7. Consider a random sample y1; y2; : : : ; yn from a distribution with probability density
function
f (y; ) = 2ye y2= for y > 0
where the parameter > 0. Find the maximum likelihood estimate of based on these observations.
2.7. CHAPTER 2 PROBLEMS 77
8. The following model has been proposed for the distribution of the number of o¤spring Y in a family, for a large population of families:
P (Y = 0; ) = 1 2 1
and
P (Y = k; ) = k for k = 1; 2; : : : and 0 < < 1 2:
(a) Suppose that n families are selected at random and that fy is the number of
families with y children (f0+ f1+ = n). Determine the maximum likelihood
estimate of .
(b) Consider a di¤erent type of sampling wherein a single child is selected at random and the size of family the child comes from is determined. Let Y represent the number of children in the family. Show that
P (Y = y; ) = cy y for y = 1; 2; : : :
and determine c.
(c) Suppose that the type of sampling in part (b) was used and that with n = 33 the following data were obtained:
y 1 2 3 4
fy 22 7 3 1
Determine the maximum likelihood estimate of . Estimate the probability a couple has no children.
(d) Suppose the sample in (c) was incorrectly assumed to have arisen from the sam- pling plan in (a). What would ^ be found to be? This problem shows that the way the data have been collected can a¤ect the model.
9. Radioactive particles are emitted randomly over time from a source at an average rate of per second. In n time periods of varying lengths t1; t2; : : : ; tn(seconds), the num-
bers of particles emitted (as determined by an automatic counter) were y1; y2; : : : ; yn
respectively.
(a) Determine an estimate of from these data. What assumptions have you made to do this?
(b) Suppose that instead of knowing the yi’s, we know only whether or not there
was one or more particles emitted in each time interval. Making a suitable assumption, give the likelihood function for based on these data, and describe how you could …nd the maximum likelihood estimate of .
78 2. STATISTICAL MODELS AND MAXIMUM LIKELIHOOD ESTIMATION
10. In a study of osteoporosis, the heights in centimeters of a sample of 351 elderly women randomly selected from a community were recorded as follows:
156 163 169 161 154 156 163 164 156 166 177 158 150 164 159 157 166 163 153 161 170 159 170 157 156 156 153 178 161 164 158 158 162 160 150 162 155 161 158 163 158 162 163 152 173 159 154 155 164 163 164 157 152 154 173 154 162 163 163 165 160 162 155 160 151 163 160 165 166 178 153 160 156 151 165 169 157 152 164 166 160 165 163 158 153 162 163 162 164 155 155 161 162 156 169 159 159 159 158 160 165 152 157 149 169 154 146 156 157 163 166 165 155 151 157 156 160 170 158 165 167 162 153 156 163 157 147 163 161 161 153 155 166 159 157 152 159 166 160 157 153 159 156 152 151 171 162 158 152 157 162 168 155 155 155 161 157 158 153 155 161 160 160 170 163 153 159 169 155 161 156 153 156 158 164 160 157 158 157 156 160 161 167 162 158 163 147 153 155 159 156 161 158 164 163 155 155 158 165 176 158 155 150 154 164 145 153 169 160 159 159 163 148 171 158 158 157 158 168 161 165 167 158 158 161 160 163 163 169 163 164 150 154 165 158 161 156 171 163 170 154 158 162 164 158 165 158 156 162 160 164 165 157 167 142 166 163 163 151 163 153 157 159 152 169 154 155 167 164 170 174 155 157 170 159 170 155 168 152 165 158 162 173 154 167 158 159 152 158 167 164 170 164 166 170 160 148 168 151 153 150 165 165 147 162 165 158 145 150 164 161 157 163 166 162 163 160 162 153 168 163 160 165 156 158 155 168 160 153 163 161 145 161 166 154 147 161 155 158 161 163 157 156 152 156 165 159 170 160 152 153
(a) Construct a frequency histogram and determine whether the data appear to be approximately Normally distributed.
(b) Determine the sample mean y and the sample standard deviation s for these data. Compare the proportion of observations in the interval [y s; y + s] and [y 2s; y + 2s] with the proportion one would expect if the data were Normally distributed with these parameters.
(c) Find the interquartile range for these data. What is the relationship between the IQR and for Normally distributed data?
(d) Find the …ve-number summary for these data.
(e) Draw a boxplot for these data. Does it resemble a boxplot for Normal data? (f) Plot a qqplot for these data and again assess whether the data are approximately
2.7. CHAPTER 2 PROBLEMS 79
11. Consider the data on heights of adult males and females from Chapter 1. (The data are posted on the course webpage.)
(a) Assuming that for each sex the heights Y in the population from which the sam- ples were drawn is adequately represented by Y G( ; ), obtain the maximum likelihood estimates ^ and ^ in each case.
(b) Give the maximum likelihood estimates for Q (0:1) and Q (0:9), the 10th and 90th percentiles of the height distribution for males and for females.
(c) Give the maximum likelihood estimate for the probability P (Y > 1:83) for males and females (i.e. the fraction of the population over 1:83 m, or 6 ft).
(d) A simpler estimate of P (Y > 1:83) that doesn’t use the Gaussian model is
number of person in sample with y > 1:83 n
where here n = 150. Obtain these estimates for males and for females. Can you think of any advantages for this estimate over the one in part (c)? Can you think of any disadvantages?
(e) Suggest and try a method of estimating the 10th and 90th percentile of the height distribution that is similar to that in part (d).
12. The lifetimes of 92 right front disc brakes pads for a speci…c car model are posted in the …le brakelife.text on the course webpage. The lifetimes y are in km driven, and correspond to the point at which the brake pads in new cars are reduced to a speci…ed thickness.
(a) Assuming a G( ; ) model for the lifetimes, determine the maximum likelihood estimates of and based on the data. How well does the Gaussian model …t the data?
(b) Another model for such data is given by
f (y; ; ) = p 1 2 yexp " 1 2 log y 2# ; for y > 0:
(Note: Show using methods you learned in your course on probability that if X v G( ; ) then Y = log X has the probability density function given above.) Using this model determine the maximum likelihood estimates of and based on the data. How well does this model …t the data? Which of the two models describes the data better?
80 2. STATISTICAL MODELS AND MAXIMUM LIKELIHOOD ESTIMATION
13. In a large population of males ages 40 - 50, the proportion who are regular smok- ers is where 0 < < 1 and the proportion who have hypertension (high blood pressure) is where 0 < < 1. If the events S (a person is a smoker) and H (a person has hypertension) are independent, then for a man picked at random from the population the probabilities he falls into the four categories SH; SH; SH; SH are respectively, ; (1 ); (1 ) ; (1 )(1 ). Explain why this is true.
(a) Suppose that 100 men are selected and the numbers in each of the four categories are as follows:
Category SH SH SH SH
Frequency 20 15 22 43
Assuming that S and H are independent events, determine the likelihood func- tion for and based on the Multinomial distribution, and …nd the maximum likelihood estimates of and .
(b) Compute the expected frequencies for each of the four categories using the max- imum likelihood estimates. Do you think the model used is appropriate? Why might it be inappropriate?
14. Censored lifetime data: Consider the Exponential distribution as a model for the lifetimes of equipment. In experiments, it is often not feasible to run the study long enough that all the pieces of equipment fail. For example, suppose that n pieces of equipment are each tested for a maximum of C hours (C is called a “censoring time”). The observed data are: k (where 0 k n) pieces fail, at times y1; : : : ; yk and n k
pieces are still working after time C.
(a) If Y has an Exponential( ) distribution, show that P (Y > C; ) = e C= , for
C > 0:
(b) Determine the likelihood function for based on the observed data described above. Show that the maximum likelihood estimate of is
^ = 1 k k P i=1 yi+ (n k)C :
(c) What does part (b) give when k = 0? Explain this intuitively.
(d) A standard test for the reliability of electronic components is to subject them to large ‡uctuations in temperature inside specially designed ovens. For one particular type of component, 50 units were tested and k = 5 failed before 400 hours, when the test was terminated, with
5
P
i=1
yi = 450 hours. Find the maximum
2.7. CHAPTER 2 PROBLEMS 81
15. Poisson model with a covariate: Let Y represent the number of claims in a given year for a single general insurance policy holder. Each policy holder has a numerical “risk score”x assigned by the company, based on available information. The risk score may be used as a covariate (explanatory variable) when modeling the distribution of Y , and it has been found that models of the form
P (Y = yjx) = [ (x)]
y
y! e
(x) for y = 0; 1; : : :
where (x) = exp( + x), are useful.
(a) Suppose that n randomly chosen policy holders with risk scores x1; x2; : : : ; xn
had y1; y2; : : : ; yn claims, respectively, in a given year. Determine the likelihood
function for and based on these data. (b) Can ^ and ^ be found explicitly?
16. Interpreting qqplots: Consider the following datasets de…ned by R commands. For each generate the Normal qqplot using qqnorm(y) and on the basis of the qq- plot determine whether the underlying distribution is symmetric, light-tailed, heavy tailed, whether the skewness is positive, negative or approximately 0, and whether the kurtosis is larger or smaller than that of the Gaussian, i.e. 3. Repeat changing the sample size n = 100 to n = 25. How much more di¢ cult is it in this case to draw a clear conclusion? (a) y<-rnorm(100) (b) y<-runif(100) (c) y<-rexp(100) (d) y<-rgamma(100,4,1) (e) y<-rt(100,3) (f) y<-rcauchy(100)
17. A Normal qqplot was generated for 100 values of a variate. See Figure 2.14. Based on this qqplot, answer the following questions:
(a) What is the approximate value of the sample median of these data? (b) What is the approximate value of the IQR of these data?
(c) Would the frequency histogram of these data be reasonably symmetric about the sample mean?
(d) The frequency histogram for these data would most resemble a Normal probabil- ity density function, an Exponential probability density function or a Uniform probability density function?
82 2. STATISTICAL MODELS AND MAXIMUM LIKELIHOOD ESTIMATION -3 -2 -1 0 1 2 3 0 0. 2 0. 4 0. 6 0. 8 1
S t andard Norm al Q uant iles
Q uant iles of I nput S am ple
Q Q P lot of S am ple Dat a vers us S t andard Norm al