Let \( X_1, \ldots, X_n \) be a random sample from a distribution with one of two pdfs. If \( \theta = 1 \), then \( f(x; \theta = 1) = I(0 < x < 1) \). If \( \theta = 2 \), then \( f(x; \theta = 2) = 2xI(0 < x < 1) \).(a) Give a general form of the MLE of \( \theta \).(b) You are given a sample data of 3 values: 0.4, 0.5, 0.8. Find the MLE estimate of \( \theta \) based on the data.

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Let X1, X2, ..., Xn be a random sample from a distribution with a pdf -ax ae ;x > 0 f (x) = 0, elsewhere. (a) Find the method of moments estimator for a. (b) Find the MLE for a. (c) Find the MLE for the variance of the distribution.
Consider a random sample of size n from a distribution with discrete pdf f(x; p) = p(1 – p)*; x = 0, 1, ..., zero otherwise. (a) Find the MLE of p. (b) Find the MLE of 8 = (1 – p)/p. (c) Find the CRLB for variances of unbiased estimators of 6. (d) Is the MLE of 0 a UMVUE? (e) Is the MLE of 8 MSE consistent? (f) Find the asymptotic distribution of the MLE of 8. (g) Let ē = nX/(n+ 1). Find risk functions of both & and X using the loss function L(t; 8) = (t – 0)*(e² + ®).
Let X1,...,Xn be a random sample from a distribution with mean μ and variance σ2. Find the value of the following quantities (as a function of μ and σ2). (a) E(X1 −X2 +X3 −X4) (b) Var(X1 − X2 + X3 − X4) (c) E(4X1 −3X2 +2X3 −X4)
Let Y, represent the ith normal population with unknown mean , and unknown variance of for i=1,2. Consider independent random samples, Ya, Y2. the ith population with sample mean Y, and sample variance S² = Yin, of size n,, from (Y-₁². (a) What is the distribution of Y,? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho : 4 = o versus Ha i Pio when of is unknown and n, is small. : (e) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.
F. Suppose that a random variable X has normal distribution with mean µ = 2 and variance σ 2 = 9, that is, X ∼ N(2, 9). (30) E[(X + 2)^2 ] is (a) 20 (b) 25 (c) 15 (d) 30 (31) The variance of X/2 + 3 is (a) 9/4 (b) 3/8 (c) 9 (d) 9/2
Let x = (x1.X2 .. .x,. . ,,) be a data set with a sample mean x. Show that E(x, – x) = 0.
Suppose you have a joint distribution of x and y where • x has population mean ug and population variance o? • y has population mean ly and population variance o, and the population covariance between them is Ery. The population Pearson correlation is then given by Ery OxJy We collect n pairs of data, (X;, Yi), i = 1,.. , n. Each pair is an independent draw from this distribution. (a) Suppose we estimate our population covariance with n 1 (x; – Ha)(yi – µy). i=1 Is this an unbiased estimator of the population covariance? Show why or why not. (b) Suppose we estimate our correlation with (xi – Ha)(Yi – Hy) - n i=1 Ox0y Is this an unbiased estimator of the population correlation? Show why or why not. =WI
Every year, the students at a school are given a musical aptitude test that rates them from 0 (no musical aptitude) to 5 (high musical aptitude). This year's results were: Aptitude Score 0 1 2 3 4 5 Frequency 4 1 1 4 4 1 The mean (T) aptitude score: The median aptitude score: (Please show your answer to 1 decimal place.) (Please separate your answers by ',' in bimodal situation. Enter DNE The mode aptitude score: if there is no mode.) Textbook Measures of Center
Q. Let the joint pmf of x and y be defined by f (x,y) = (x+y)/32 where x = 1,2 and y = 1,2,3,4
Let Y, represent the ith normal population with unknown mean #, and unknown variance of for i=1,2. Consider independent random samples, Ya, Ya, Yin, of size n,, from the ith population with sample mean Y, and sample variance S² = (₁-P) ². (d) Define V₁, a function of S1, that has a chi-square distribution. (e) Is V₁ a pivotal function to find a confidence interval for o?? Explain with argument. (f) Find a (1-a) x 100% confidence interval for of
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