here for 109 variables; expected std error =sqrt(9/109)=0.2873
therefore P(Xbar>16.09)=1-P(X<bar<16.09)=1-P(Z<(16.09-16)/0.2873)=1-P(Z<0.31)=1-0.62
therefore d =0.62
(1 point) Let X1, , X109, be IID with mean 16 and variance 9. By the...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below we describe a ? test for the null and alternative hypotheses We divide the sample space into 5 disjoint subsets refered to as bins A1(-00,-2), A2 -(-2,-0.5), As -(-0.5,0.5), A4 (0.5,2) As -(2, oo). as functions of X, by Now, define discrete random variables For example, if Xi --0.1, then Xi є Аз and so Y;-3. In other words, Y, is the label of...
2. Approximate the following probabilities for sufficiently large n by applying CLT. Please provide answers using the standard CDF Φ(z) = P(Z 2) where Z ~ N(0,1), instead of real numbers. (a) For X Binomial(n, 1/4), P(X/Vn 0.5). (b) For X1, , , , Xn iid Uniform(0, 1), PK, < 2/(3 ) (c) For Y ~ χ2(n), P(Y < n).
1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that, U, can be approximated by the N (np, np(1-P) distribution, for large n and fixed <p<1. 2. Suppose that X1, X3, X. are iid N ( 0°). Where and a both assumed to be unknown. Let @ -( a). Find jointly sufficient statistics for .
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
Suppose that X1, X2,....Xn is an iid sample of size n from a Pareto pdf of the form 0-1) otherwise, where θ > 0. (a) Find θ the method of moments (MOM) estimator for θ For what values of θ does θ exist? Why? (b) Find θ, the maximum likelihood estimator (MLE) for θ. (c) Show explicitly that the MLE depends on the sufficient statistic for this Pareto family but that the MOM estimator does not
1. The random variables Xi, X2,... are independent and identically distributed (iid), . .. are independent and identica each with pdf f given in Assignment 4, Question 1. Let s, X1 + . .. + Xn. Using the Central Limit Theorem and the graph of the standard normal distribution in Figure 1, approximate the probability P(S100 > 600). Express your answer in the format x.x - 10*. Verify your answer by simulating 10,000 outcomes of S1o0 and counting how many...
Suppose X1, X2, .., Xn is an iid sample from where >0. (a) Derive the size α likelihood ratio test (LRT) for Ho : θ-Bo versus H : θ θο. Derive the power function of the LRT (b) Suppose that n 10, Derive the most powerful (MP) level α-0.10 test of Ho : θ 1 versus Hi: 0-2. Calculate the power of your test
1. Let X1, ..., Xn be random sample from a distribution with mean y and variance o2 < 0. Prove that E[S] So, where S denotes sample standard deviation. 10 points
3. Suppose X1, X2, , Xn are iid based on the random variable modeled by 2,0-1 (1-2)a-1 where 0 < x < 1 and α > 0 a. Find an equation that the MLE for a must satisfy. Note: You will not be able to explicitly solve for the MLE as in other problems b. If you are told E(X) = 2 and Var(X) = 8a14, example where someone might prefer the MME over the MLE find the MME for...
5.13. Suppose X1, X2, , xn are iid N(μ, σ2), where-oo < μ < 00 and σ2 > 0. (a) Consider the statistic cS2, where c is a constant and S2 is the usual sample variance (denominator -n-1). Find the value of c that minimizes 2112 var(cS2 (b) Consider the normal subfamily where σ2-112, where μ > 0. Let S denote the sample standard deviation. Find a linear combination cl O2 , whose expectation is equal to μ. Find the...