o Read: Handouts 4 8.: 5 0 Supplemental Reading in Devore book: Chapters 1 85 4 0 Submit for grading the following problems: P1. ( 10 Points) Let Y have a double exponential distribution, that is, Y has pdf and cdf in the following form with parameters 6, 6 > 0: 167(6)?) for y Modules: AssignSBrainSize BRAINSIZE - SMALL LITTER SIZE 0.42 0.86 0.88 1.11 1.34 1.38 1.42 1.47 1.63 1.73 2.17 2.42 2.48 2.74 2.74 2.79 2.90 3.12 3.18 3.27 3.30 3.61 3.63 4.13 4.40 5.00 5.20 5.59 7.04 7.15 7.25 7.75 8.00 8.84 9.30 9.68 10.32 10.41 10.48 11.29 12.30 12.53 12.69 14.14 14.15 14.27 14.56 15.84 18.55 19.73 20.00 BRAINSIZE - LARGE LITTER SIZE 0.94 1.26 1.44 1.49 1.63 1.80 2.00 2.00 2.56 2.58 3.24 3.39 3.53 3.77 4.36 4.41 4.60 4.67 5.39 6.25 7.02 7.89 7.97 8.00 8.28 8.83 8.91 8.96 9.92 11.36 12.15 14.40 16.00 18.61 18.75 19.05 21.00 21.41 23.27 24.71 25.00 28.75 30.23 35.45 A software package uses the estimator @(u) : Y((n_1)u+1) as the estimator of Q(u). 0 Calculate the estimates of the Quartiles: of Q(.25), (2(5), Q(.75) for just the Large Litter Size using the software package's formula. P3. ( 26 points) Using the data from Problem 2 for just the Large Litter Size, we want to estimate the pdf f (y) for the relative brain weights of the 44 species of mammal. The kernel density estimate of f (y) is given by A \" Yi men2.2149 h ) Suppose we use the Gaussian kernel: KW) : e'l/Q and a bandwidth of h : 3. ( a.) Estimate f (3) and f (16) using the kernel density estimator. ( b.) Using a relative frequency histogram with bin width of 5, estimate the values of f (3) and f(15)- ( c.) Which data value provides the smallest contribution to the kernel density estimator at y:16, A f (16)? ( (1.) Which data value provides the largest contribution to the kernel density estimator at y=16, A f(16)? P4. ( 30 points ) Using the relative Brain Weight data, answer the following questions: ( a.) Produce the following plots of the data: estimates of the pdf, cdf, and quantile function for both Small and Large litter sizes. ( b.) Describe the underlying distribution of the relative brain weights for both Small and Large litter sizes. ( c.) Based on the graphs, what are your conclusions about the relationship between litter size and relative brain weights? P5. ( 24 Points) Select the letter of the best answer for each question. No explanation is needed for your selection. 1. The function which provides the most detailed description for the realizations of a random variable is A. B. C. D. E. the probability density (mass) function, pdf f () the quantile function, Q(-) the survival function, 30 the cumulative distribution function, cdf F(-) all the above functions are equivalent 2. A relative frequency histogram having classes of greatly dilferent class widths was used as an estimator of a continuous population pdf. The relative frequency was plotted versus the class intervals. This plot will not be an appropriate estimator of the population pdf because A. B. C. D. E. all the intervals are not the same width. the relative frequency varies greatly by class width. the area under the curve is not proportional to one. the area under the curve for each class is not an estimator of the probability of that class. In fact it is an unbiased estimator of the pdf. 3. A relative frequency histogram having classes of greatly dilferent class widths was used as an estimator of a continuous population pdf. The relative frequency was plotted versus the class intervals. The plot will result in a graphical distortion. The plot can be corrected by A. B. C. D. E. making all the intervals have the same width. plotting the relative frequency divided by class width. making sure that the area under the curve adds to one increasing the sample size. In fact there will not be a distortion since it is an unbiased estimator of the pdf. 4. A kernel density estimator was used as an estimator of a continuous population pdf, f (y) The kernel density estimator is generally a vastly improved estimator over a density histogram (plot of N13: H vs Class 2') because A. in using the histogram, it is necessary to select the number of bins, bin widths, and their location. B. there are too many spurious modes using the histogram C. the area under the curve adds to 1 for the kernel density estimator. D. the kernel density estimator makes use of all the data in estimating f (y) whereas the histogram only uses those data values in the same bin as y. E. Answers A, C, and D are all correct. 5. In using a kernel density estimator to estimate a population pdf based on a random sample Y1, - - - ,Ym the design factor which is least crucial in determining the effectiveness of the estimator is A. the sample size, n B. the number of plotting points. m provided m > 50 C. the bandwidth, h D. the kernel k(-) E. all four factors are equally crucial 6. A kernel density estimator is an estimator of a population pdf, f (y) The bandwidth of the kernel density estimator is selected by A. using the uniform distribution to randomly select a value between 0 and 1. B. taking the value which minimizes the asymptotic integrated mean square error. C. taking the value which produces a curve having area closest to 1. D. taking the value of the bandwidth which yields maximum entropy. E. asking Dr. Sheather. 7. A random sample of 11 data values is obtained from a process having an absolutely continuous cdf of unknown shape. The metallurgist wants to select the best tting distribution amongst several candidate cdfs. She decides to select the distribution which has mean and variance most closely matching the corresponding sample mean and variance. The major weakness in this approach is A. the mean and variance may be highly inated by outliers B. the empirical distribution function contains more information about the tails of the distribution than does the mean and variance she should have used robust estimators of the location and scale parameters there are many distributions having the same mean and variance but very different shapes 91.5.0 the moments of a distribution determine the distribution, hence there is no weakness in the approach 8. The median is a trimmed mean with level of trimming equal to A. B. C. D. E. 0% 25% 50% 75% none of the above 9. The standard deviation is preferred to MAD as a measure of population dispersion when the population distribution 10. 11. 12. A. B. C. D. E. has absolutely no outliers. has a skewed but shorttailed distribution. has a lognormal distribution. has a normal distribution. cannot be determined with the given information. The coefcient of kurtosis A. B. C. D. measures the dispersion of the distribution about two values p :l: 0. measures the peakedness of a distribution. measures the concentration of the mass of a distribution in the tails of the distribution. all of the above Alternatives to a for measuring the dispersion in a distribution are SI QR and MAD. Which of the following statements about these measures are TRUE? A. 5.59?\" Both measures are equal if the pdf for the distribution is skewed. SI QR is preferred to MAD if the distribution has very heavy tails For the normal distribution, SIQR is preferred to MAD all of the above none of the above A government study of the average monthly nitrate levels in the Mississippi river, Ni, just prior to its entry into the Gulf of Mexico is modeled as N; = 22.3 +.6Nt71 + 6.4, where egs are iid r.v.s, E[et] = 0, Var[et] = 2.8, egs are independent of Nt's The mean and variance of Nt are given by A. ECO?\" p 2 22.3, 0'2 : 2.8 p, = 55.75, 02 = 4.375 p : 34.84, 0'2 : 2.8 p : 22.3, 0'2 : 4.375 The values of ,u and a2 would change from month to month