![Question 6 [2 marks] Let X1, X2, ..., X, be a random sample from the Poisson distribution with mean e. a. Express the VAR,(Xi](http://img.homeworklib.com/questions/bbe3b000-9aef-11ea-a976-e7b24c205409.png?x-oss-process=image/resize,w_560)
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![Now dlogh d - hot sxi Logo - E logai! do -nt exi = 9 > ô = Sxi M u noun E tô) ( x 2 () - - [80] - no o Flence o is unbiased.](http://img.homeworklib.com/questions/ce1dfb60-a23a-11ea-9de1-b300b652db25.png?x-oss-process=image/resize,w_560)
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Question 6 [2 marks] Let X1, X2, ..., X, be a random sample from...
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability fun...
Advanced Statistics, I need help with (c) and (d)
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the inte...
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
7. (10 points) Let X1, X, be a random sample of size 2) from a Poisson...
7. (10 points) Let X1, X, be a random sample of size 2) from a Poisson distribution with mean = 1, and assume Xand X, are independent. Let Y = min{X1, X2}, then P(Y = 1) = P(X1 =1nx, > 1) + P(X2 = 1n Xi > 1) - P(X1 = 1n X2 = 1) = 2e-1-3e-2 (a) Show that P(X1 = 1n X2 = 1) = -2 (b) Show that P(X1 = 1n X2 > 1) = -1-e-?
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the...
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0 unbiased estimator of e. estimator eCYis an (c) Get the lower bound for the variance of the unbiased estimator found in (b)
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0...
7.77. If X1, X2,.., X, is a random sample from a distribution with p.d.f. f(x;0)=0*xe-, 0 <x< 00, zero elsewhe...
7.77. If X1, X2,.., X, is a random sample from a distribution with p.d.f. f(x;0)=0*xe-, 0 <x< 00, zero elsewhere, where 0 e< ao: (a) Find the m.l.e., 6. of 0. Is 6 unbiased? X and then compute E(0). Hint: First find the p.d.f. of Y = (b) Argue that Y is a complete sufficient statistic for 8. (c) Find the unbiased minimum variance estimator of 0. (d) Show that X/Y and Y are (e) What is the distribution of...
Please let me know how to solve 7.6.5. 6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with p...
Please let me know how to solve 7.6.5.
6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with parameter θ > 0. (a) Find the MVUE of P(X < 1)-(1 +0)c". Hint: Let u(x)-1, where Y = Σ1Xi. 1, zero elsewhere, and find Elu(Xi)|Y = y, xỉ (b) Express the MVUE as a function of the mle of θ. (c) Determine the asymptotic distribution of the mle of θ (d) Obtain the mle of P(X...
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution....
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
1. Let X1, X2,...,x. be a random sample from the unif(0,0) distribution (a) Find an unbiased...
1. Let X1, X2,...,x. be a random sample from the unif(0,0) distribution (a) Find an unbiased estimatior of O based on the sample mean X (b) Find an unbiased estimator of based on the sample maximum X (c) Which estimator is better in terms of variance?
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
Advanced Statistics, I need help with (c) and (d)
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
7. (10 points) Let X1, X, be a random sample of size 2) from a Poisson distribution with mean = 1, and assume Xand X, are independent. Let Y = min{X1, X2}, then P(Y = 1) = P(X1 =1nx, > 1) + P(X2 = 1n Xi > 1) - P(X1 = 1n X2 = 1) = 2e-1-3e-2 (a) Show that P(X1 = 1n X2 = 1) = -2 (b) Show that P(X1 = 1n X2 > 1) = -1-e-?
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0 unbiased estimator of e. estimator eCYis an (c) Get the lower bound for the variance of the unbiased estimator found in (b)
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0...
7.77. If X1, X2,.., X, is a random sample from a distribution with p.d.f. f(x;0)=0*xe-, 0 <x< 00, zero elsewhere, where 0 e< ao: (a) Find the m.l.e., 6. of 0. Is 6 unbiased? X and then compute E(0). Hint: First find the p.d.f. of Y = (b) Argue that Y is a complete sufficient statistic for 8. (c) Find the unbiased minimum variance estimator of 0. (d) Show that X/Y and Y are (e) What is the distribution of...
Please let me know how to solve 7.6.5.
6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with parameter θ > 0. (a) Find the MVUE of P(X < 1)-(1 +0)c". Hint: Let u(x)-1, where Y = Σ1Xi. 1, zero elsewhere, and find Elu(Xi)|Y = y, xỉ (b) Express the MVUE as a function of the mle of θ. (c) Determine the asymptotic distribution of the mle of θ (d) Obtain the mle of P(X...
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
1. Let X1, X2,...,x. be a random sample from the unif(0,0) distribution (a) Find an unbiased estimatior of O based on the sample mean X (b) Find an unbiased estimator of based on the sample maximum X (c) Which estimator is better in terms of variance?
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
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