Problem III. (12 points) Consider the following probability distribution. X 0 2 4 6 P(X =...
Problem III. (12 points) Consider the following probability distribution. X 0 6 P(X = x) 1/4 1/4 1/4 1/4 1. (2 points) Find E(X). 2. (5 points) Find the sampling distribution of the sample mean X for samples of size = 2. n = 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as @(X1, ... , Xn) = -XI(X; #0, and X: #6), п i=1 where...
Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4 1/4 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as n B(x,,,X,) X;I(Xi 70, and X; #6), n i=1 where I(-) is an indicator function, I(X; # 0, and Xi # 6) = 0, if X; € {0,6}, and I(X; # 0, and X; + 6) = 1, if Xi...
Problem III. (12 points) Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4 1/4 1. (2 points) Find E(X). 2. (5 points) Find the sampling distribution of the sample mean à for samples of size n = 2.
4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwise, , where 0 > 0. Let X,i,.., X, be a random sample from this distribution. Not cavered 2011 (a) [6 marks] Find-4MM, the nethod of -moment estimator for θ for θ? If not, construct-an unbiased'estimator forg based on b) 8 marks Let X(n) n unbia estimator MM. CMM inbiase ( = max(X,, , Xn). Let 0- be another estimator of θ. 18θ...
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...
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where θ is an unknown parameter, θ > 0. Let X, be the sample mean and X(n)max(Xi, X2,,Xn). (i) Show that θ¡n-(1 + )Xn ls an unbiased estimator of θ. Find its mean square error and check whether θ¡r, is consistent for θ. (i) Show that nX(n) is a consistent estimator of o (ii) Assume n > 1 and find MSE's of 02n, and compare...
, xn is an iid sample from fx(x10)-θe-8z1(x > 0), where θ > 0. Suppose X1, X2, For n 2 2, n- is the uniformly minimum variance unbiased estimator (UMVUE) of 0 (d) For this part only, suppose that n-1. If T(Xi) is an unbiased estimator of e, show that Pe(T(X) 0)>0
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.
Let X1, X2, ..., Xn be a random sample with probability density function a) Is ˜θ unbiased for θ? Explain. b) Is ˜θ consistent for θ? Explain. c) Find the limiting distribution of √ n( ˜θ − θ). need only C,D, and E Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...