1. An article in the Journal of the American Statistical Association ["Markov Chain Monte Carlo Methods for Computing Bayes Factors: A Comparative Review" (2001, Vol. 96, pp. 1122-1132)] analyzed the tabulated data on compressive strength parallel to the grain versus resin-adjusted density for specimens of radiata pine. The data are shown below. Compressive Strength Density Compressive Strength Density 3040 29.2 3840 30.7 2470 24.7 3800 32.7 3610 32.3 4600 32.6 3480 31.3 1900 22.1 3810 31.5 2530 25.3 2330 24.5 2920 30.8 1800 19.9 4990 38.9
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- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.27. An article in Radio Engineering and Electronic Physics (1980, Vol. 25, pp. 74-79) investigated the behavior of a stochastic generator in the presence of external noise. The number of periods was measured in a sample of 100 trains for each of two different levels of noise voltage, 100 and 150 mV. For 100 mV, the mean number of periods in a train was 7.9 with s1 = 2.6. For 150 mV, the mean was 6.9 with s2 = 2.4. Use α = 0.01 and assume that each population is normally distributed and the two population variances are equal. (a) It was originally suspected that raising noise voltage would reduce mean number of periods. Do the data support this claim? (b) Calculate a confidence interval to answer the question in part (a).What are the difficulties in estimating the following model? Use as much detail as possible in answering this question while considering the Gauss-Markov assumptions and OLS estimator. Economic productivity = β0 + β1Unemployment + β2Innovation + θiControls + ei Where unemployment is the average unemployment rate of a country and innovation is an index of R&D performance.
- A study of pine nut crops in the American southwest from 1940 to 1947 hypothised that nut production followed a Markov chain. The data suggested that if one year's crop was good, then the probabilities that the following year's crop would be good, fair, or poor were 0.08, 0.07, and 0.85, respectively; if one year's crop was fair, then the probabilities that the follow-ing year's crop would be good, fair, or poor were 0.09, 0.11, and 0.80, respectively; if one year's crop was poor, then the probabilities that the following year's crop would be good, fair, or poor were 0.11, 0.05, and 0.84, respectively. (a) Write down the transition matrix for this Markov chain. (b) If the pine nut crop was good in 1940, find the probabilities of a good crop in the years 1941 through 1945. (c) In the long run, what proportion of the crops will be good, fair, and poorBased on the data that has been gathered what is the probability of disasters in July & Augustus using Markov chain ?1- The number of items produced in a factory during a week is known to be a randomvariable with mean 50● Using Markov's inequality, what can you say about the probability that this week'sproduction exceeds 75?● If the variance of one week's production is equal to 25, then using Chebyshev'sinequality, what can be said about the probability that this week's production isbetween 40 and 60?
- In a Bernoulli process with success rate .4, if we consider the total number of successes X which happened on trials 7,9,15,16, 23, and if Y is the total number of successes on trials 15,23, 32,38, 43, then what is the covariance of X with Y?Reference: Camm J.D. et al., (2019) Business Analytics: descriptive, predictive and prescriptive (3rd edition). Cengage ____________________________________________________________________________________ Jay Gatsby categorizes wines into one of three clusters. The centroids of these clusters, describing the average characteristics of a wine in each cluster, are listed in the following table. Characteristic Cluster 1 Cluster 2 Cluster 3 Alcohol 0.819 0.164 -0.937 MalicAcid -0.329 0.869 -0.368 Ash 0.248 0.186 -0.393 Alcalinity…In the study of evolutionary behavior, the Trivers-Willard hypothesis indicates that healthyparents should tend to have more male offspring than female, and that weaker parents should tendto have more female offspring than male. This tendency may maximize the number of each parent’sgrandchildren (and thus help to ensure that its genetic code is preserved) since a healthy maleoffspring can win many mates, but a relatively unhealthy offspring has the best chance of mating ifit is female. In an experiment to examine this hypothesis, a group of 40 opossums were monitoredand 20 of them were given an enhanced diet. After a certain period of time, the opossums with theenhanced diet had raised 19 male offspring and 14 female offspring, and the opossums without theenhanced diet had raised 15 male offspring and 15 female offspring. Does this finding provideevidence in support of the Trivers-Willard hypothesis? (a) Describe the unknown parameters ?1 and ?2. Then state the null and…
- Suppose that the index model for two Canadian stocks HD and ML is estimated with the following results: RHD =-0.03+2.10RM+eHD R-squared =0.7 RML =0.06+1.60RM+eML R-squared =0.6 σM =0.15 where M is S&P/TSX Comp Index and RX is the excess return of stock X. What is the covariance and the correlation coefficient between HD and ML? For portfolio P with investment proportion of 0.4 in HD and 0.6 in ML, calculate the systematic risk, non-systematic risk, and total risk of P.Show what correction (using the generalized least squares method) we should use if the form of heteroskedasticity was the following, show mathematically For both a) and b)In a continuous-time surplus model, the claim severity is distributed as BN(2, 0.4). Determine the Lundberg upper bound for the probability of ultimate ruin if the initial surplus is 2 and the prIn a continuous-time surplus model, the claim severity is distributed as BN(2, 0.4). Determine the Lundberg upper bound for the probability of ultimate ruin if the initial surplus is 2 and the premium loading factor is 0.25.emium loading factor is 0.25.