Exercise 3.7 Suppose X, are iid Exp(1). Find the limiting distribution of Va(X-X()-1), wh ere Xo...
Suppose that Y1,Y2,··· ,Yn is an iid from Y ∼ U(0,3). Find the limiting distribution of ¯ Y . What is the probability of average of Y from a random sample of 10 that exceed 1.6?
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
Let Xi....,Xn,..., ~iid Exp(1) and let Yn) be the sample maximum of the first n observations. Show that the limiting distribution of Zn-(Y(n)-log n) has CDF F(z) exp{-e-*), z є R.
8(100) Let X1,,Xn be iid from r(a, 6). (1)(50) Find the limiting distribution of the MLE of B. (2)(30) Find the limiting distribution of the MLE of B when a is known. (3)(20) Compare two asymptotic variances in (1) and (2), and make comment on it. 1ラ 8(100) Let X1,,Xn be iid from r(a, 6). (1)(50) Find the limiting distribution of the MLE of B. (2)(30) Find the limiting distribution of the MLE of B when a is known. (3)(20)...
Part 1: Derive the expected value and find the asymptotic distribution. Part 2: Find the consistent estimator and use the central limit theorem b. Derive the expected value of X for the Weibull(X,2) distribution. c. Suppose X,.. .X,~iid Uniffo,0). Find the asymptotic distribution of Z-n(-Xm) max Let X, X, ~İ.id. Exp(β). a. Find a consistent estimator for the second moment E(X"). Use the mgf of X to prove that your estimator is consistent in the case β=2 b. Use the...
40. Suppose that X,.. , X N(0,o?). (a) Determine the asymptotic distribution of the iid re ciprocal of the second sample moment, that is, of h-n/Σ2lX. (b) Find a variance stabilizing transformation for the statistic (1/n) EX2
3. Find a closed formula for the exponential generating function A(x) Σ an,n wh n+1-(n + 1)(m-n + 1), a,-1. ere an satisty the recursion a 3. Find a closed formula for the exponential generating function A(x) Σ an,n wh n+1-(n + 1)(m-n + 1), a,-1. ere an satisty the recursion a
6. (10 points) Suppose X ~ Exp(1) and Y = -ln(X) (a) Find the cumulative distribution function of Y. (b) Find the probability density function of Y.
3. Again, let XXn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xo) and X(a) (b) Define R-X(n) - Xu) as the sample range. Find the pdf of R (c) It turns out, if Xi, X, n(iid) Uniform(0,e), E(R)- What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively.
7.2.10 Suppose that X, .., X, are iid with the Rayleigh distribution, that is the common pdf is where θ(> 0) is the unknown parameter. Find the MLE for θ, is the MLE sufficient for θ?