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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2) Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function Where, ß > 0. If fram e ¹- {** f(x;0)=B for x > 4 otherwise "1 g) What is the maximum likelihood estimator of g (B) = 2(ß + 1). h) Is the maximum likelihood estimator of g (B) = 2(B + 1) unbiased or not?
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.
Please help me for Question 2
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
7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
10. Two random variables X and Y take on the values i and 2 with probability 1/2' (i = 1,2, ...). Show that the probabilities sum to one. Find the expected value of X and Y.
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
7. Suppose that the joint p.d.f. of two random variables X and Y is as follows: (4 – 2x – y) for x > 0, y > 0, f(x, y) = and 2x + y 2|X = 0.5).
Example 17.37. (a) Let x1, x2, ..., Xn be a random sample from the uniform distribution with p.d.f. : 1,0 0 f(x, 0) 0, elsewhere Obtain the maximum likelihood estimator for 0.

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