Homework Answers
TOPIC: Finding the required distribution.
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically...
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically distributed i.id. N(14x, σ2) random variables, and let Yı, . . . , Yrn denote an independent sequence of iid. Nụy, σ2) ran- dom variables. il Λί and Y is an unbiased estimator of μ for any value of λ in the unit interval, i.e. 0 < λ < 1. 2. Verify that the variance of this estimator is minimised when and determine the...
Suppose X1,X2, ,Xm are iid exponential with mean A. Suppose Yı,Yo, exponential with mean β2-Suppo...
Suppose X1,X2, ,Xm are iid exponential with mean A. Suppose Yı,Yo, exponential with mean β2-Suppose the samples are independent. , Yn are iid (a) Derive the likelihood ratio test (LRT) statistic λ(x,y) for testing versus and show that it is a function of ti-ti (x)-Σ-iz; and t2-t2(y)-Σ1Uj. (b) Show how you could perform a size a test in part (a) using the F distribution
Suppose X1,X2, ,Xm are iid exponential with mean A. Suppose Yı,Yo, exponential with mean β2-Suppose the...
and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ-...
and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ- ΣΧ be the sample mean, S2 be the sample variance. j-1 n-1 Σ--Σ( - 1' -nΣΤ-β). (i) Prove that Using it, determine the distribution of X (ii) Find the m.g.f. of X. n ΣT- ) Σ- 7 7 n (iii) Indicate the distributions ofJ 2 , respectively. and (iii) Given that X and S are independent, derive the m.g.f of (n-15, and then, σ'...
Let X and Y be independent exponentially distribution random variables with rate α and β respectively....
Let X and Y be independent exponentially distribution
random variables with rate α and β respectively. Find P (X > Y
).
Question 13: Let X and Y be independent exponentially distribution random variables with rate a and B respectively. Find P(X> Y).
5. Let 11,D, , , ,Zn and yı, y2, . . . , ym denote independent...
5. Let 11,D, , , ,Zn and yı, y2, . . . , ym denote independent observed random samples of size n and m taken from two normally distributed populations with the same mean μ but different variances σ and σ . lihood estimator for the common mean μ based on the combined sample Find the maximum like . Is pmle unbiased? Find the variance of nle. - Define the following estimator n+ m Is μ unbiased. Find the variance...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution,...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
Let X1, X2, . . . , Xn be a random sample of size n from...
Let X1, X2, . . . , Xn be a random sample of size n from a normal population with mean µX and variance σ ^2 . Let Y1, Y2, . . . , Ym be a random sample of size m from a normal population with mean µY and variance σ ^2 . Also, assume that these two random samples are independent. It is desired to test the following hypotheses H0 : σX = σY versus H1 : σX...
2. Consider a large population with mean μ and known standard deviation σ = 5. There are two inde...
2. Consider a large population with mean μ and known standard deviation σ = 5. There are two independent simple random samples of this population, one with n 150, and the other with n2 = 400, Denote the two sample means by , and X2, respectively. Let Cli and C12 be the usual 95% confidence intervals, constructed from each of the two samples. What is the probability that at the same time, X E CI2 and X2 E CI?
2....
QUESTION 5 Suppose that Yı, Y2,.., Yn independent variables such that where β is an unknown...
QUESTION 5 Suppose that Yı, Y2,.., Yn independent variables such that where β is an unknown parameter, X1, x2-.., xn are known real numbers, and el,e2 independent random errors each with a normal distribution with mean 0 and variance ơ2 ,en are (a) Show that is an unbiased estimator of β. What is the variance of the estimator? (b) Given that the probability density function of Y is elsewhere, show that the maximum likelihood estimator of β is not the...
U means Uniform distribution 2. Let X be a r.v. distributed as U(α, β). Show that...
U means Uniform distribution
2. Let X be a r.v. distributed as U(α, β). Show that its ch. f. and m.g.f.x and Mr, respectively, are given by and IM x it(β-a) , t(B-a) ii) By differentiating (ax, show that E(X)-(α + β) / 2 and T 2 (X)-(α-β)2 / 12.
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically distributed i.id. N(14x, σ2) random variables, and let Yı, . . . , Yrn denote an independent sequence of iid. Nụy, σ2) ran- dom variables. il Λί and Y is an unbiased estimator of μ for any value of λ in the unit interval, i.e. 0 < λ < 1. 2. Verify that the variance of this estimator is minimised when and determine the...
Suppose X1,X2, ,Xm are iid exponential with mean A. Suppose Yı,Yo, exponential with mean β2-Suppose the samples are independent. , Yn are iid (a) Derive the likelihood ratio test (LRT) statistic λ(x,y) for testing versus and show that it is a function of ti-ti (x)-Σ-iz; and t2-t2(y)-Σ1Uj. (b) Show how you could perform a size a test in part (a) using the F distribution
Suppose X1,X2, ,Xm are iid exponential with mean A. Suppose Yı,Yo, exponential with mean β2-Suppose the...
and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ- ΣΧ be the sample mean, S2 be the sample variance. j-1 n-1 Σ--Σ( - 1' -nΣΤ-β). (i) Prove that Using it, determine the distribution of X (ii) Find the m.g.f. of X. n ΣT- ) Σ- 7 7 n (iii) Indicate the distributions ofJ 2 , respectively. and (iii) Given that X and S are independent, derive the m.g.f of (n-15, and then, σ'...
Let X and Y be independent exponentially distribution
random variables with rate α and β respectively. Find P (X > Y
).
Question 13: Let X and Y be independent exponentially distribution random variables with rate a and B respectively. Find P(X> Y).
5. Let 11,D, , , ,Zn and yı, y2, . . . , ym denote independent observed random samples of size n and m taken from two normally distributed populations with the same mean μ but different variances σ and σ . lihood estimator for the common mean μ based on the combined sample Find the maximum like . Is pmle unbiased? Find the variance of nle. - Define the following estimator n+ m Is μ unbiased. Find the variance...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
2. Consider a large population with mean μ and known standard deviation σ = 5. There are two independent simple random samples of this population, one with n 150, and the other with n2 = 400, Denote the two sample means by , and X2, respectively. Let Cli and C12 be the usual 95% confidence intervals, constructed from each of the two samples. What is the probability that at the same time, X E CI2 and X2 E CI?
2....
QUESTION 5 Suppose that Yı, Y2,.., Yn independent variables such that where β is an unknown parameter, X1, x2-.., xn are known real numbers, and el,e2 independent random errors each with a normal distribution with mean 0 and variance ơ2 ,en are (a) Show that is an unbiased estimator of β. What is the variance of the estimator? (b) Given that the probability density function of Y is elsewhere, show that the maximum likelihood estimator of β is not the...
U means Uniform distribution
2. Let X be a r.v. distributed as U(α, β). Show that its ch. f. and m.g.f.x and Mr, respectively, are given by and IM x it(β-a) , t(B-a) ii) By differentiating (ax, show that E(X)-(α + β) / 2 and T 2 (X)-(α-β)2 / 12.
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