b) Find the method of moments estimator for 0.c) Show that the estimator is unbiased.d) Show that the estimator is consistent.e) Show that the estimator is efficient.9106 Q3.Let X,X2, X be a random sample of size n from a Poisson population with2**.distribution function f(x: 0)%3Dfor x 0, 1,2,...xtShow that X X, is a sufficient estimator for 8. [Hint Use factorization theorem)

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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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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a) Let X₁, X₂, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) you and Y = -√n. VR i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal rando... variable. iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X₁, X₂, X, and Y₁, Y₂.... Y be random samples from populations with moment generating functions Mx, (t) = eat+t² and My (t) = (-), respectively. 25 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ X3 + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)2> 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-Hy ≥10) <0.04? QUESTION A
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
1. Let y1,.. , Yn be a random sample from a uniform distribution on the interval (0, B). Let B1 2 Li=1 Yi and B, = ) Max(y1,.., Yn) be estimators for B. Find the sampling distribution of B2 Show that B, and B, are unbiased estimators for B. 11.
4. Let X,,X2, .. ,X„ be a random sample form a uniform distribution, f(X), = 0*(1– 0)*-X; 0 < x < 1. Assume T, = X, and T, = (HEx. Check the unbiasedness property for T, and T2. Find the MSE of T, and T2. Can you preferer one of them? Discuss your reasons. a. b. с.
50. Let X₁, X2, ..., X50 be i.i.d. with E(X) = 25 and V(X₂) = 10. Consider the random variable defined by 50 X₁ the sample average X = Σ Apply the Central Limit Theorem to answer the following: n i=1 (a) What is the approximate distribution of X (include parameter(s))? (b) Approximate P(X> 25.5). (c) Approximate P(|X-25 <0.8). (d) Approximate the 0.975 quantile (97.5th%tile) of X.
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b) Let X₁, X₂, X, and Y₁, Y₂,...,Y be random samples from populations with moment generating 25 functions Mx, (t) = e³t+t² and My(t) = respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂-X₂ + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|-|≥10) < 0.04?

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Q3. distribution function f (x: 0)- Let X,, X2, ,X, be a random sample of size n from a Poisson population with for x 0, 1, 2., ... a) Show that X X is a sufficient estimator for 8. [Hint Use factorization theorem OY