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Questions and Answers of Statistics Econometrics

20.2 Use CERN’s boosted decision tree program, and the data given at the website https://archive.ics.uci.edu/ml/datasets/MiniBooNE+particle+identification to separate out signal (νe events) from
20.1 Prove the relation, σ (z) = σ (z)(1 − σ (z)) that was used in the discussion of the cross-entropy cost function.
19.2 When the 1987 Supernova (SN1987A) occurred, neutrino interactions from neutrinos made in the supernova were seen by the IMB experiment and by the Kamiokande experiment.Most of the events in the
19.1 Verify that Eq. 19.2, does follow from Eq. 19.1, if K(x) = F(x). Hint: Look at the equations for sums of integers given as a hint in Exercise 13.3.
18.2 Generate a Monte Carlo distribution of 1000 events where one-fourth of the events are background distributed according to 1.0 + 0.3E for 0 ≤ E ≤ 2, one-half of the events have a
18.1 Consider the function f (z) = sin(z). Divide this into eight equal intervals, (nine knots) from z = 0 to z = π. Use the cubic B-spline method to interpolate between the nine values of z and
17.3 We make n measurements of x from a normal distribution and estimate the variance using the biased estimate, variance = (xi − xAV)2/n, where xAV = xi /n. For n = 5 and n = 10, generate
17.2 Find the Bartlett S function for the case that one makes n measurements of the time of an event on a distribution whose frequency function varies with time as e−t/τ from t = 0 to t = T and is
17.1 Suppose we wish to show that a coin is reasonably fair. To this end, we will try to distinguish between the hypotheses (a) The probability of a toss of the coin yielding heads is 0.5. (b) The
16.2 Show that (16.40), is the solution for (16.38) Hint: Add and subtract x. Then expand the terms and evaluate the cross term using (16.36)
16.1 Show that (16.36), is the solution for ∂χ2(x) ∂ x x=x f = 0
15.3 Try the above exercise, first taking the file saved in Exercise 15.2 and binning the data into a 40 bin histogram. Then use the χ2 method with a general minimizing program. Although some
15.2 You will again need a general minimizing program for this exercise. In Exercise 8.1, you worked out a method for generating Monte Carlo events with a Breit– Wigner (B-W) distribution in the
15.1 The decay rate of orthopositronium has been measured by atomic physicists. Gidley and Nico (1991) have kindly provided me with a set of data in which they measured the number of decays in
14.3 Show that if the probability of obtaining n events is given by the Poisson distribution with mean θ and the probability of θ is given by the gamma distribution with mean α/μ, and variance
14.2 Popcorn is popping in a hot pan and a measurement made of how high each kernel pops. In a given batch, 25 kernels are observed to pop more than 4in. high. If the total number of kernels is
14.1 Derive and plot the symmetrical confidence limit curves shown in Fig. 14.6, for the binomial distribution for the case n = 20. (If you use a computer, you might as well have it plot the whole
13.4 For the toy model in Sect. 13.2, we had a naive solution for the estimate of the number of events in (13.7) and a solution in which the variance was defined as x; rather than the theoretically
13.3 A set of n thin wire proportional chambers uniformly spaced at positions along the x axis has made a series of measurements of track position, i.e., x, y values for a straight track. We wish to
13.2 Imagine that we measure n pairs of x, y values. Suppose y is a normally dis- tributed variable with constant variance o but with a mean that depends linearly on x. m = a +(x-x). where X is the
13.1 Suppose we have a visible detector such as a bubble chamber with a fast beam of short-lived particles of known energy passing through it. We might imagine, for instance, that we have a beam of
11.1 Show that the Lindeberg criterion is satisfied for the binomial distribution and, therefore, the binomial distribution does approach the normal distribution for large n. (Let each xi be a single
10.2 Suppose we have a collection of diatomic molecules that we excite with laser beam pulses. The molecules can ionize giving up an electron or dissociate giving up an ion. At each pulse, we measure
10.1 Consider a two-dimensional density function proportional to exp −(2x 2 + 2x + 1/2 + 2x y + 8y2 ) . (a) Find m1, m2, σ1, σ2, ρ, λi j . (b) Write the correctly normalized density
9.5 Derive Eq. 9.17 for the mean and variance for the number of failures preceding the mth success.
9.4 For Sect. 9.1, Item 5, repair people servicing machines, calculate for m = 3, n = 20, R = 0.1 a table for k = 1 − 10, showing the values for the number of machines being serviced, waiting to
9.3 Derive Eq. 9.14. Hint: Consider the equilibrium situation with k machines needing service for three cases, k = 0, 1 ≤ k ≤ m, and k > m. Show these equilibrium relations are satisfied by (k +
9.2 Show that the equilibrium equations, Eqs. 9.9 and 9.10 satisfy the equilibrium versions of Eq. 9.8.
9.1 Show that for the infinite buffer problem at the beginning of this chapter, E{k} = R/(1 − R) and var{k} = R/(1 − R)2.
8.5 Let us consider an example of the importance sampling variant of the acceptance rejection method. Consider again the problem of generating normally distributed pseudorandom numbers for x > 0.
8.4 In this exercise, we will examine the efficiency of detection of events in an experiment. You will generate 300 events of D0 decays and see if each event is detectable. To be detectable, the D0
8.3 From this point on, a number of the exercises will ask you to draw graphs and make fits to data. Both kinds of these problems ask you to calculate a set of data, say x vs y and then operate on
8.2 Devise a fast way to generate a Poisson distribution using Monte Carlo methods assuming λ is less than 20, using random numbers uniformly distributed between 0 and 1. (Hint: Consider using a
8.1 A distribution of events as a function of energy is expected to have a Breit–Wigner resonance (M,) and a background density function proportional to 1/(a + bE)3. It is expected that the number
7.7 Find the mean and standard deviation of theχ2 distribution using the characteristic functions.
7.6 Prove Theorem 7.8 for generating functions (Eq. 7.25). Q(x) ≡ ∞ k=0 qk xk = P (x) P (0) , where qk = (k + 1)pk+1/ < k >.
7.5 Prove Eq. 7.22. h(x) = ∞ j=0 π(j)xj = ∞ j=0 xj ∞ k1=0 ··· ∞ km=0 δ(j, i ki) m i=1 p(i) ki = m i=1 Pi (x); where Pi (x) = ∞ k=0 p(i) k xk . The probability π(j) was π(j) =
7.4 Prove Eq. 7.20.
7.3 Prove the generating functions Eqs. 7.14, 7.18.
7.2 Suppose, for X , a discrete random variable, fn = Pr(X = n) has generating function F(s). Let gn = Pr(X > n) and have generating function G(s). Show that G(s) = (1 − F(s))/(1 − s).
7.1 If xi is a set of independent measurements, let xAV = xi/n. Prove that for the Breit–Wigner distribution the distribution of xAV is the same as that of xi. (Recall that for a normal
6.5 Suppose we have N books numbered 1 to N on a shelf and randomly select two of them located at j1 and j2. The probability for the sum j1 + j2 is called the triangle distribution. The distribution
6.4 We have seen that the binomial distribution approaches the normal distribution for large n. Compare the exact calculation of the probability of having 20 heads out of 40 tosses of a fair coin
6.3 Suppose we make five measurements of a quantity we know to be normally distributed with mean 0.05. We obtain 0.041, 0.064, 0.055, 0.046, 0.060. Estimate the variance of the distribution. Suppose
6.2 You are measuring a quantity and are uncertain of the error. You make three measurements obtaining 1.0, 2.0, 1.0. Estimate the variance. The next day you come back to your apparatus and measure
6.1 Suppose the probability that an electronic component fails to function in the time interval (t, t + dt) is φ(t) dt provided the component has functioned properly up to time t. Given ∞ 0
5.5 This problem has arisen in evaluating random backgrounds in an experiment involving many counting channels. Suppose one has three channels and each of the three channels is excited randomly and
5.4 Suppose we are measuring second harmonic generation when pulsing a laser on a solid. However, the detector we have available is only a threshold detector and the time resolution is such that it
5.3 We consider collisions of a high-energy proton with a proton at rest. Imagine the resultant products consist of the two protons and some number of pions. Suppose we imagine the number of pions is
5.2 A nuclear physics experiment is running. It requires 50 independent counters sensitive to ionizing radiation. The counters are checked between data runs and are observed to have a 1% chance of
5.1 The negative binomial distribution gives the probability of having the mth success occur on the rth Bernoulli trial, where p is the probability of the occurrence of the success of a single trial.
4.2 Assume that we start with r red and b black balls in an urn. We randomly draw balls out one at a time without looking into the urn. (a) We draw them out and place them to one side of the urn (no
4.1 Consider the probability of players having complete suits, A,...,K in bridge hands in a given round. (a) Find the probability that one particular player has a complete suit. (b) Find the
3.6 Two measurements for κ, the compressibility of liquid He, gave 3.87 ± 0.04 × 10−12 and 3.95 ± 0.05 × 10−12 cm2/dyn. From these two experiments, find the best estimate and its error for
3.5 Atomic physicists have measured parity violation effects in a series of delicate experiments. In one kind of experiment, they prepare two samples identically except for reversed external fields
3.4 Suppose that we are measuring counting rates with a scintillator and that we are measuring a signal and a background. First we measure a background plus a signal, then a background only,
3.3 Suppose one has measured a counting rate in a scintillation counter twice, obtaining R1 = m1/T1 and R2 = m2/T2, respectively, as results. m1, m2 are the number of counts obtained and T1, T2 are
3.2 Imagine that a charged particle is passing through matter undergoing multiple scattering. Neglect energy loss, i.e., let θ 2 y = Kx. We wish to estimate the direction of motion of the particle
3.1 Suppose we obtain n independent results xi from a distribution F(x). Let xAV be the mean of these n values, the sample mean. Define σ2 s ≡ 1 n − 1 n i = 1 (xi − xAV) 2 . (3.41) Show that
2.4 A coin is tossed until for the first time the same result appears twice in succession. To every possible pattern of n tosses, attribute probability 1/2". Describe the sample space. Find the
2.3 Suppose one has two independent radioactive sources. Disintegrations from each are counted by separate scintillators with expected input rates N and N2. Ignore the dead times of the array.
2.2 Let ,, = x" be the nth moment of a distribution, , be the nth central moment of the distribution, and m = be the mean of the distribution. Show that 3=3-3m2+2m.
2.1 A 1-cm-long matchstick is dropped randomly onto a piece of paper with parallel lines spaced 2 cm apart marked on it. You observe the fraction of the time that a matchstick intersects a line. Show
11-4 The BALANCE investigators who conducted the study of drug treatments for bipolar people in Problem 5-9 also collected data on when experimental subjects had emergent mood episodes for 36 months.
11-3 What is the sample size for each experimental group to obtain .80 power using a log rank test to detect a significant difference (with a = .05) in steady state survival rates between .40 and .20?
11-2 Taking care of old people on an outpatient basis is less costly than caring for them in nursing homes or hospitals, but health professionals have expressed concern about how well it is possible
11-1 Surgery is an accepted therapeutic approach for treating cancer patients with metastases in their lungs.Philippe Girard and colleagues†collected data on 35 people who had metastases removed
10-8 In his continuing effort to become famous, the author of an introductory biostatistics text invented a new way to test if some treatment changes an individual’s response.Each experimental
10-7 Rework Problem 9-1 using the Wilcoxon signed-rank test.
10-5 To determine whether or not offspring of parents with type II diabetes have abnormal glucose levels compared to offspring without a parental history of type II diabetes. Gerald Berenson and
10-4 Rework Problem 9-6 using methods based on ranks.
10-3 Rework Problem 9-5 using methods based on ranks.
10-2 The inappropriate and overuse of antibiotics is a well-recognized problem in medicine. To test whether it was possible to encourage more appropriate use of antibiotics in elderly hospitalized
10-1 Despite progresses in technique, adhesions (the abnormal connection between tissues inside the body formed during healing following surgery) continue to be a problem in abdominal surgery, such
9-10 Review all original articles published in the New England Journal of Medicine during the last 12 months. How many of these articles present the results of experiments that should be analyzed
9-9 The data in Problem 9-8 could also be presented in the following form as shown in Table 9-13. How would these data be analyzed? If this result differs from the analysis in Problem 9-9, explain
9-8 In the fetus, there is a connection between the aorta and the artery going to the lungs called the ductus arteriosus that permits the heart to bypass the nonfunctioning lungs and circulate blood
9-7 What is the power of the test in Problem 9-7 to find a 100 mL change in food intake with 95% confidence?
9-6 In general, levels of the hormone testosterone decrease during periods of stress. Because physical and psychological stressors are inevitable in the life of soldiers, the military is very
9-5 In addition to measuring FEV1 (the experiment described in conjunction with Fig. 9-6), Michel and colleagues took measurements of immune response in their subjects, including measuring the amount
9-4 Rework Problem 9-2 as a repeated measures analysis of variance. What is the arithmetic relationship between F and t ?
9-3 What are the chances of detecting a halving of the heart rate variability in Problem 9-2 with 95% confidence?Note that the power chart in Figure 6-9 also applies to the paired t test.
9-2 Secondhand tobacco smoke increases the risk of a heart attack. In order to investigate the mechanisms for this effect, C. Arden Pope III and his colleagues†studied whether breathing secondhand
9-1 Several epidemiological studies have shown that people who have a diet high in flavenols (which are in tea, wine, cocoa products and various fruits) have lower rates of dying from coronary artery
8-11 Clinical and epidemiologic studies have demonstrated an association between high blood pressure, diabetes, and high levels of lipids measured in blood. In addition, several studies demonstrated
8-10 What sample size is necessary to have an 80% power for detecting a correlation between journal circulation and selectivity with 95% confidence if the actual correlation in the population is .6?
8-9 What is the power of the study of journal circulation and selectivity described in Figure 8-12 to detect a correlation of .6 with 95% confidence? (There are 113 journals in the sample.)
8-8 As part of a study of the nature of cancers of the gum and lower jaw, Eiji Nakayama and his colleagues†were interested in relating the extent of cancer invasion, determined by direct
8-7 Erectile dysfunction is widely recognized to be associated with diabetes and cardiovascular disease. To investigate whether erectile dysfunction was associated with lower urinary tract
8-6 Arteries adjust their size on a minute-to-minute basis to meet the needs of the body for blood to carry oxygen to the tissues and to remove waste products. A substantial part of this response is
8-5 The ability to measure hormone levels based on a blood spot (like that used by diabetics for glucose monitoring) has several advantages over measurements based on a blood draw. First, blood spots
8-4 Polychlorinated biphenyls (PCBs) are compounds that were once used as an insulating material in electrical transformers before being banned in the United States during the 1970s because of
8-3 The plots in Box 8-3 show data from four different experiments together with the associated observations.Compute the regression and correlation coefficients for each of these four sets of data.
8-2 Plot the data and compute the linear regression of Y on X and correlation coefficient for each of the sets of observations shown in Table 8-10. In each case, draw the regression line on the same
8-1 Plot the data and compute the linear regression of Y on X and correlation coefficient for each of the sets of observations shown in Table 8-9. In each case, draw the regression line on the same
7-11 Rework Problem 5-14 using confidence intervals
7-10 Rework Problem 5-13 using confidence intervals.
7-9 Rework Problem 5-12 using confidence intervals.
7-8 Rework Problem 5-11 using confidence intervals.
7-7 Rework Problem 5-5 using confidence intervals.

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