2. Consider the negative binomial distribution with parameters r,p and having pmfnb(x;r,p) =Ꮖ(* + r − ¹) p*(1 − p)²p'(1-p) x = 0, 1, 2, 3, … ….(a) Supposer 2, then show thatT-1p =X+r−1is an unbiased estimator for p. (Hint: write out E(p), then cancel out x+r −1 insidethe sum).(b) A reporter wishing to interview five individuals who support a certain candidate (forpresidency?) begins asking people whether they support (S) or not support (F) the can-didate.If they observe the following sequence of responses SFFSfffffffffffSSS, esti-mate p the true proportion of people who support the candidate.How does the estimate change if the following sequence of responses were observedssssfffffffffffffs.Does it matter to the estimate when the first four S's appear in the sequence of responses?

Answered: Image /qna-images/answer/b72007b1-a2d4-4e02-8926-39e11a9abd22.jpg

Answered: 2. Consider the negative binomial…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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5. Suppose that X₁, X2,..., Xn are an iid sample from the distribution given below, f(x) = (1/2)(1+X0); -1 < X < 1. Show that 3X is a consistent estimator of 0.
7. Suppose that X is a random variable for which the m.g.f. is as follows: V (1) = (3e' + e) for -0
7. Let (r1, 12 ., T„) be a random sample from a normal distribution with mean u and a variance o?. Also let S (x; - )² be the sum of squares of the random sample. %3D State without proof the sampling distribution of the following; (a) t(x) = ao + b;a;, i = 0,1, .., n, a, are positive real constants. i=l n(ī – µ)² (x – 4) (b) t(r) = g2 (f) t(x) = (c) t(x) = (x – 4) (g) t(x) = i=1 (c) t(x)= i=1 i=1 (d) t(r) = E (*; – äm-n)² %3D j=1 i=m+1 (h) t(x) = m - 1 n - m - 1 (x – 4) (e) t(r) =
2. Suppose the distribution of Y given X = x is a normal random variable with mean x and variance x². And X follows an uniform distribution on (0,1) (a) Find E(XY) (b) Let U = X, V = Y/X. Show that U and V are independent
4. Suppose X1,... , Xn is a random sample of size n drawn from a Poisson pdf where A is an unknown parameter. Show X X is an unbiased estimator for X
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
5. Find the moment generating function of the exponentially distributed random vari- able with pdf Xe-, x≥0 and use it to evaluate the mean and variance of X.
Let Y, represent the ith normal population with unknown mean 4, 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 S?=₁1(Y-₁². (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - Y₂ can be used to test the equality of the two population means, #₁1 and 12, when o=o=o² is known. (i) Define appropriate rejection regions, in terms of Y₁-Y₂, for testing Ho: #₁ = 1₂ against a two-sided alternative hypothesis at the a level of significance.
Suppose that f(x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.
Suppose that f (x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.
3 For an SRS {4₁};=, from the normal distribution. ==// (Y-M) 2 fy (y;u, 0²) - 1 е värvoze Show that the sample mean I is a consistent estimator,
10. True or false. (i) –1 < Cov(X, Y) <1 for any two random variables X, Y. (ii) When two random variables have zero correlation they must be independent. (iii) Cov(13.X, 13X) = 13Var(X). Those statements that you mark as false, explain why and give a corrected version.

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