7.7.4 As in Problem 7.6.3, let X₁,..., X be a set of independent random variables with a U(0, 0)distribution, and letT= max{X₁,..., X₂}Explain why the likelihood function L(x₁,..., Xn, 0) is equal toif 0 > t = max {x,...,xn), and is equal to 0 otherwise.Sketch the likelihood function against 0, and deduce that the maximum likelihood estimate of 0 is Ô =t. What is the bias of this point estimate?

A random sample X1,...., Xn of size 'n' is taken from a distribution U0,θ. The PDF of X is given by:…

Answered: 4 As in Problem 7.6.3, let X₁,..., X be…

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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Question 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.
Question 2 i=1,...,n. Assume that y Let y; be the number of transactions made by customer i in a year, Poisson (2), 2 > 0 is unknown, i = 1,...,n. ~ (a) What is the likelihood function, p(y | λ), for 2 provided by the data y = (y₁,...,yn)? (b) Define the Jeffreys prior, p,(0), for a scalar parameter 0, and derive its form from the Poisson likelihood for this example. (c) Assuming the Jeffreys prior, p,(e), for 0, what is the posterior distribution for ? Suppose now we use a Gamma distribution, Gamma(a, b), as a prior for 2 where a > 0 and b 0 are known. (d) What is the posterior distribution, p(λ | y), for λ under the Poisson model and the prior Gamma(a, b) distribution. (e) Suppose ab. For an uninformative prior, do we need a large or small value for a? If we choose a small value for a, what is the effect on the posterior distribution for compared to a large a?
3. If x1, X2, X3, ..., Xn is a random sample from a normal 1 distribution N (µ, 1) show that t= E xí is an unbiased estimator of n i = 1 H² + 1.
2. The random variable X is the amount of fructose in apples (in ounces) grown at a local orchard and is represented by the following cdf: 2(x +), 1
2. Let X1, ..., Xn Bernoulli(p), with n = 20 and p unknown. Let Y = X1+ ...+ X20. (a) What is the likelihood function of p conditional on the n = 20 Bernoulli observations? 4 (b) What is the likelihood function of p conditional on y? (c) True or False, and briefly explain: Both likelihood functions result in the same MLE of p. Answer this question without finding the MLE in each case. (d) Consider the sample proportion of the Bernoulli variables, p=X. Show that Var(p) = p(1-p)/n.
Please answer all
4. Consider a random sample {X¡, i = 1, 2, ...n} of size n from a distribution with pdf f(x; 8) = 0(1+ x)-(1+0) if x > 0, and 0 otherwise. (a) Show that the maximum likelihood estimator (mle) ô of 0 is a function of E-1 In(1 + X¡) (b) Let Y = In (1+X), find pdf of Y and mgf of Y. %3D (c) Using result (b) to show that Z = E, In(1 + X;) has a Gamma distribution. Specify the parameters of the gamma distribution (hint: find mgf of Z) (d) For test Ho: 0=1 vs Hi : 0 > 1, we reject null hypothesis if (X1, ..., Xn) € C, where C= {(X, .., Xn), E-, In(1 + X;) > k}. Give an expression to obtain k for given significance level a and what is the expression of power when 0 =1 (no need to integrate the function).

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