1. On Homework 1, you worked with a random sample of size n from a d...
Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = (1/θ)e^(−x/θ) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ. Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = θe^(−θx) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ.
1. Consider a random sample of size n from a population with pdf: f (x) = (1 -p-p, 0 <p<1, x= 1, 2, ... (a) Show that converges in probability to p. (b) Show that converges in probability to p (1 – p). (c) Find the limiting distribution of
8. Consider a random sample of size n from a distribution with pdf f(x) = 0 else (a) Find the pdf of the smallest order statistic, X(i) b) Find E() and Var(X)) c) Find the pdf of the largest order statistic, X(n)
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.
Consider a random sample of size n from the distribution with pdf (In )* f(x; 0) = { 0.c! -, 10, =0,1,... otherwise where 0 > 0. (a) (10 pts) Find a complete sufficient statistic for 0. (b) (10 pts) Using Lehmann-Scheffe theorem, find the UMVUE of Ine. You may need the identity c=
4. (6 marks) Consider a random sample of size n from a distribution with pdf f(x:0) 26-1 if 0 1 and zero otherwise; θ 0, Find the UMVUE of 1/θ x
Let X, denote the mean of a random sample of size n from a distribution that has pdf (9xe-3x, x>0 f(x) = 0, otherwise Let Yn = mn (Ăn – ). Find the limit distribution of O N(0, 1) O N(0, 0) O N(o, ž) O N(0, 3) other
Let X, X,, ..., X, denote a random sample of size n from a population with pdf (10) = b exp(@m()).0<x<1 where (<O<0. Derive that the likelihood ratio test of H.:0=1 versus H, :0 #1 in terms of T(x) = ŽI (3)
A random sample of size n = 2 is taken from the p.d.f f(x) = {1 for 0 ≤ x ≤ 1 and 0 otherwise. Find P(X-bar ≥ 0.9) 3. A random sample of size n = 2 is taken from the p.d.f 1 for 0 < x < 1 f(30 0 otherwise. Find P(X > 0.9)
Consider a random sample of size n from a distribution with pdf (In O* S(x; 6) = Ox! x = 0, 1, ...;0 > 1 10 otherwise (a) Find a complete sufficient statistic for 8. (b) Find the MLE of O. (c) Find the CRLB for 6 (d) Find the UMVUE of In e. (e) Find the UMVUE of (In )? (1) Find the CRLB for (In 02