(Mathematical Statistics) Problem 5. Let we have a sample of size n from the pdf f(x|0)...
Problem 5 Let Y1 denote the minimum of a random sample of size n from a distribution that has pdf f(x) e(,0x< o0, zero elsewhere X- n (Y1 0), find the cumulative distribution function (cdf) for Zn = n (Y1 - 0), and Let Zn find the limiting cdf of Zn as n >oo.
difficult…… 2and4 thanks Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ. (b) Find an...
4. Let f(x) = 22xe-2x,x>> 0). Assume that we have a random sample of size n from this distribution. Find the maximum likelihood estimator of 2.
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x < oo. (a) Find the method of moments estimator of θ. (b) Find the maxinum likelihood estimator of θ
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE. Let Xi , X2,. … X, denote a random sample of size n...
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise, where θ > 0. a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ (c) Find the pdf of θ in (b).
5. Let X1,...,Xn be a random sample from the pdf f(\) = 6x-2 where 0 <O<< 0. (a) Find the MLE of e. You need to justify it is a local maximum. (b) Find the method of moments estimator of 0.
(5) Let X, i = 1,...,n be iid sample from density fx(x) = f(x) e-/201(x > 0), 4 > 0 V TO (a) Find k. (b) Find E(X). (c) Find Var(X). (d) Find the MLE for 0. (e) Find MOM estimator for A. (f) Find bias for MLE. (g) Find MSE of MLE. (h) Let Y = x, find probability density function of Y. (i) Let Y = X?, find cumulative distribution function of Y. 5
Let X,X,, X, be a random sample of size 3 from a uniform distribution having pdf /(x:0) = θ,0 < x < 0,0 < θ, and let):く,), be the corresponding order statistics. a. Show that 2Y, is an unbiased estimator of 0 and find its variance. b. Y is a sufficient statistic for 8. Determine the mean and variance of Y c. Determine the joint pdf of Y, and Y,, and use it to find the conditional expectation Find the...