3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have th...
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have the joint probability function where x = (zi, 2 2:23), θ = (θί, θ2), n = x1 + 2 2 + x3, θι, θ2 > 0 and 1-0,-26, > 0. (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks] Find that the Fisher information matrix I(0) (c) [4 marks] Show that θ is an MVUE. (d)...
3. [20 marks Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and Xs have the joint probability function (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks Find that the Fisher information matrix I(0). (c) [4 marks] Show that θ is an MVUE. (d) 4 marks Find the approximate distribution of Y 2X-X2, when the sample size n is large (e) [4 marks] Assume that X-(253, 234, 513). Find the...
20 marksConsider the multinomial distribution with 3 categories, where the random variables X1,X2 and X have the joint probability function 123 [4 marks] Find the approximate distribution of Y = 2X1-X2, when the sample size n is large. 20 marksConsider the multinomial distribution with 3 categories, where the random variables X1,X2 and X have the joint probability function 123 [4 marks] Find the approximate distribution of Y = 2X1-X2, when the sample size n is large.
[20 marks] Let xi, . . . , Xn be a random sample drawn independently from a one-parameter curved normal distribution which has density -oo 〈 x 〈 oo, θ > 0, 2πθ nx, and r2 - enote T-1 Tn (d) [3 marks] Find the maximum likelihood estimator θ2 of. (You do not need to perform the second derivative test.) (e) 3 marks Find the Fisher information T( (f) [3 marks] Is θ2 an MVUE of θ? Justify your answer....
2. 20 marks] Let z1,., xn be a random sample drawn independently from a one-parameter curved normal distribution which has density -oo < x < 00, θ>0, , riid i.e., X r, and 2,2-1 Γη (e) 3 marks Find the Fisher information Z(0) (f) [3 marks] Is θ2 an MVUE of θ? Justify your answer (g) 3 marks] Assume that T = 1.32 and x-3.76 for a random sample of size n = 100. Find the Wald 95% confidence interval...
Question 5. [10 Marks] Suppose . . . ,X, be an SRS from a uniform distribution between θ and 0. a) Į1 Mark] Find the moments estimator (ME) θί of θ. b) [1 Markl Let Y- min(X1,... ,Xn) and its pdf is as follows. -ny"-1 for ye(θ,0); for y E (6,0), -, 0, -, fy(y) otherwise. Show that the maximum likelihood estimator (MLE) θ2 = ntly of θ is unbiased. c) [4 Marks] which one of θ1 and θ2 is...
Description for Question 7. A multinomial distribution for three nonnegative counts X1, X2, X3 has joint pdf given by 23 P(X1 = X1, X2 = 22, X3 = x3) (21.3.2.) pi? pºp3", X1 X2 X3 where pi + P2 + P3 1. For genotypes AA, Aa, and aa, the Hardy-Weinberg model puts the respective genotype proportions in the population at (1 - 0)?, 20(1 – 0), and 02, where 0 is the gene frequency of gene type "a" (0 <...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3). 3. (25 pts.) Let X1,...
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2 e _ θ:/z1(z > 0), where θί 〉 0, for i = 1, 2, , n. (a) Derive the form of the likelihood ratio test (LRT) statistic for testing versuS H1: not Ho. You do not have to find the distribution of the likelihood ratio test (LRT) statistic under Ho- Just find the form of the statistic. (b) From your result in part (a),...
5. Suppose that three random variables Xi, X2, and X3 have a continuous joint distribution with the following p.d.f. (x1+2x2+3z3) and f(1, r2, 3) 0 otherwise. (a) Determine the value of the constant c; (b) Find the marginal joint p.d.f. of Xi and X3; (c) Find P(Xi < 1|X2-2, X3-1)