9. Let X1,..., Xn be a sample from a population with density fo(x) a(8)h(x) if 0, < x < O2 0 otherwise where h(x) > 0, 6 = (01,02) with –∞ < O, < Oz < ∞, and a(0) = S h(x)dx is assumed to exist. Find a two-dimensional sufficient statistic for this problem and apply your result to the U 01,02] family of distributions.
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- y 2 4 1 ГО.1 0.157 P(X, Y) = x3 0.2 0.3 %3D 5 Lo.1 0.15] a) Evaluate the marginal distribution of X and Y. b) Find P(Y/X) and P(X/Y). c) Find P(Y=2/X=3).If X is an exponential random variables with rate 1, then its distribution function is given by F(x) = 1 − e−x Show that x = − ln(1 − u).3) Find the mean and variance of the random variable X with probability function or density f(x). (a)f(x)={k(1-x), –15xs1 O otherwise (b) ƒ(x) = k/3× , x = 1,2,3,... Note: in this problem, you need to first find the values of k.
- 9. Let X1, ... , Xn be a sample from a population with density fo (x) a(€)h(x) if 0, 0, 6 = (01,02) with -∞ < ®1 < Oz < ∞, and a(0) = So h(x)dx is assumed to exist. Find a two-dimensional sufficient statistic for this problem and apply your result to the U01,02) family of distributions. (01, 02) with -∞ < O1 < O2 < ∞, and a(0) %3D1. X is a normally distributed variable with mean μ = 30 and standard deviation o = 4. Find a) P(x 21) c) P(3013. Suppose a continuous random variable x has the probability density Sk(1+ x), 0 1.4) using distribution function, determine the probabilities that (d) x is less than 2.2 (e) between 1.4 and 1.6 (f) Calculate mean and variance for the probability density function.3. Suppose Y is a random sample of size 1 from a population with density function Oyº-1 0 0.5. (b) The hypotheses are Ho : 0 = 0, versus Ha : 0 > 00. What is the value of 00 if the signficance level of the test specified in (a) is 0.10? (c) Suppose a random sample of n = 7 observations is taken from this population to test the hypotheses Ho : 0 = 2 versus Ha : 0 > 2. As a decision rule, the experimenter plans to record X, the number of that exceed 0.9, and reject if X > 4. That is, the experimenter plans to use the test: ys 1 X >4 $(X) = { 0 x < 4 What is the probability of making a Type I error? Need Your Help with Part - (c)5. Suppose that Y₁, Y2, Y3 denote a random sample from an exponential distribution with density function 1 f(y) == e 6, y> 0, 00 Y₁ + 2Y₂ Five estimators are proposed for the parameter 0, Ô₁ = Y₁, 0₂: Y₁+Y₂ 03 2 = Ô₁ = Y(1) = " 3 and Ô5 = Y (a) Compute the bias of each estimator and hence identify the unbiased estimators. (b) Find for each of the estimators, the mean square error and hence the standard error. (c) Find the efficiency of 85 relative to each of the other four estimators and comment on your results.5. You have a random sample Y₁. Y₂ of size 2 from the distribution with density function 20-y f(y|0) = for ∞33.-b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x; 0) = 1 -20₁ V2πθ e 0, if - ∞0 < x < 0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.5. Suppose that the random variable X follows an exponential distribution with probability density function S(z) = de-A";0 0). Define a new random variable Y = x. (a) Show that the cumulative density function of Y is given by Fy(y) = {-exp(-Ay"; and hence, or otherwise, find the probability density function of Y. (b) Find an expression for the maximum likelihood estimator of the parameter A, using a sample Y1, 92, , Yn, from the distribution of Y. v20Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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