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- Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Derive an asymptotic distribution for the MOM estimator ˜θ = 2X¯ of the form.(b) From this choose an approximate pivot and interval to get an interval that asymptotically has100(1 − α)% coverage.Does a distribution exist for which:Mx(t) = (t)/(t-1)for |t| < 1? If yes, find it, otherwise prove that is not possibleLet {Xt : t > 0} be a compound Poisson process of parameter λ. Suppose that each of the summands of this process is constant equal to k ∈ IN . Find the distribution of Xt.
- Suppose that x has a hypergeometric distribution with N = 300, n = 30, and K=100 a. Find p(x=10) by using the hypergeometric distribution b. Find p(x=10) by using the approximate binomial distribution c. Is the binomial approximation reasonable?Prove that the rv Y=(tan^-1(X1/X2)) and Z=(X1^2+X2^2)^0.5 are independent. No distribution is given.Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample? I asked this question earlier today, but didn't quite understand all of the response. P(y1<=yn)p(y2<=yn) and so on was used, but shouldn't the yn be listed first in the inequality since we want to know if yn is the smallest?
- Use Simpson's 3/8 rule to compute the approximated solution where f (x) = 1/Cosx + 4 defined at the interval (1,4.5) and n = 77. Assume that X₁,..., Xn is a random sample from a Bernoulli (p) and let Yn n 1/1 X₁. For p # 1/2, Determine the asymptotic distribution of √n[Yn(1 – Yn) - p(1 - p)]. 7. Assume that \( X_1, ..., X_n \) is a random sample from a Bernoulli(p) and let \( {Y}_n = \1/n\sum_{i=1}^{n} X_i \). For \( p =/ 1/2), Determine the asymptotic distribution of \( \sqrt{n}({Y}_n(1 - {Y}_n) - p(1 - p)) \).Assume that your computer completes a 5000 equation back substitution in 0.005 seconds. Use the approximate operation counts n2 for back substitution and 2n3/3 for elimination to estimate how long it will take to do a complete Gaussian Elimination of this size. Round your answer to the nearest second.
- A rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u(x,0) = 8 sin(px/ 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?Theorem 6.4 states that the moment-generating function of the gamma distribution is given by Mx(t) = (1-βt)^(-α).