Exercise 5.4 Show that, if X follows the GEV distribution with parameters - § < 0, µ, and 0 > 0, then Y = 1+ (x − μ)/0 follows the Weibull distribution defined in Section 4.4.2, and identify the parameters of the standard Weibull distribution.
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![Exercise 5.4 Show that, if X follows the GEV distribution with parameters
§ < 0, μ, and > 0, then Y = 1+§ (x-μ)/0 follows the Weibull distribution
defined in Section 4.4.2, and identify the parameters of the standard Weibull
distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6d71514-91da-4fc8-8cc7-b04e6b6f461a%2Fb45f6b2a-bf84-42bc-a97c-9be5a2feb03a%2Fopy6eg_processed.jpeg&w=3840&q=75)
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- 2.2, Once an individual has been infected with a certain disease, let X represent the time (days) that elapses before the individual becomes infectious. An article proposes a Weibull distribution with a = B = 1.3, and y = 0.5. [Hint: The two-parameter Weibull distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in the equation below, %3D -xª – le-(x/B)ª Ba f(x; а, В) x 0 by x > y.] (a) Calculate P(1 1.5). (Round your answer to four decimal places.) (c) What is the 90th percentile of the distribution? (Round your answer to three decimal places.) days (d) What are the mean and standard deviation of X? (Round your answers to three decimal places.) mean days standard deviation daysLet X1, X2,...Xn be a random sample from a Poisson distribution with variance 0. Find a sufficient estimator for 0 and an unbiased estimator for 0 based on the sufficient estimator.call Wn3 a. Let X have a Poisson distribution with a mean of 3. Define Y = (X-3) ^2 , specify pdf from Y.b. Let Y have a Poisson distribution with a mean of 5. Define Y=Z+X where Z and X are independent, where X is a random variable at point a, determine the distribution and pdf of Z.
- Consider data 5,1,3,5,5,4,3,2. Poisson distibbution and want to test H0:u=u0 with u0=2. Use wilks theorem to approximate the distribution of -2logA(y) and write the pvalue for testing H0.11) Emissions of nitrogen oxides, which are major constituents of smog, can be modeled using a normal distribution. Let x denote the amount of this pollutant emitted (in parts per billion) by a randomly selected vehicle. Suppose the distribution of x can be described by a normal distribution with u = 1.8 and o = 0.5. a) A city wants to offer some sort of incentive to get the worst polluters off the road. What emission levels constitute the worst 10% of the vehicles? (Round your answer to three decimal places.)We can combine the autoregressive model with the moving average model to form an ARMA model. Here is an example. Suppose uz is stationary with mean zero, e is mean zero, variance o, and each draw is iid. Suppose we have serial correlation so that: Ut = pUt_1 + et + Oe,-1 a. Find the cov (ut, et) as function of the parameters and o. b. Find the cov (ut, ut-1) as a function of the parameters, o and o. c. Find the var(ut) as a function of the parameters and o̟. d. Now find cov(ut, ut-1) again, but this time write it as only a function of the parameters and o?.
- A report in LTO stated that the average age of taxis in the Philippines is 9 years. An operations manager of a large taxi company selects a sample of 40 taxis and finds the average age of the taxis is 8.2 years. The σ of the population is 2.3 years. At ? = 0.05, can it be concluded that the average age of the taxis in his company is less than the national average?Genetic Creutzfeldt-Jakob Disease (gCJD) is a very rare disease, and the number of deaths caused by gCJD each year in the United Kingdom can be modeled using a Poisson distribution with unknown parameter 0. Numbers of gCJD deaths for the period 2005-9 are given in Table 12. A Gamma(11, 2) was selected as a prior for 0. Year Number of deaths Table 12 2005 2006 13 9 2007 2008 2009 9 5 8 (a) State the posterior for parameter 0, given the data and prior above. (b) Calculate the posterior mode for 8, giving your answer to 2 decimal places. (c) Figure 11 shows the prior, likelihood and posterior for 8. Figure 11 10 Prior Likelihood Posterior 15 Comment on the relative contributions of the prior and data to the posterior, justifying your opinion. How have the data affected opinion about ?The random variable X has a Bernoulli distribution with parameter p. A random sampleX1, X2, . . . , Xn of size n is taken of X. Show that the sample proportionX1 + X2 + · · · + Xnnis a minimum variance unbiased estimator of p.Given that X1, X2, . . . , Xn forms a random sample of size n from a geometric population withparameter p, show thatY =n∑j=1
- Assume I observe 3 data points x1, x2, and x3 drawn independently from anunknown distribution. Given a model M, I can calculate the likelihood for each data point as Pr(x1 | M) = 0.5, Pr(x2 | M) = 0.1, and Pr(x3 | M) = 0.2. What is thelikelihood of seeing all of these data points, given the model M: Pr(x1, x2, x3 | M)?Consider data 5,1,3,5,5,4,3,2 of the Poisson distribution with expectation u. We want to test H0:u=u0 with u0=2. Write the log-likelihood for the data under H0 and the likelihood ratio A(y) for testing H0Prob. 3 Let X be a random variable with cumulative distribution function (cdf) given by (1-e-x², x ≥ 0 ={1,- x<0 Find the probability that the random variable X falls within one standard deviation of its mean. Fx (x) =
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