Let X1 and X2 be independent random variables with mean μ and variance σ2. Suppose we...
Let x and x, be independent random variables with Mean u and variance o2. Suppose that we have two estimators Of u : A @= X1 + X2 2 and ©2 = X, +3X2 2 (a) Are both estimators unbiased estimators of u? (b) What is the variance of each estimator?
Suppose that X1, X2 are two independent random variables with a common mean μ, but two different variances σ12 > σ22. Consider the family of estimators Wα = αX1 + (1−α)X2 where 0 ≤ α ≤ 1 (a) Show that Wα is an unbiased estimator of μ for any value of α. (b) Find the value of α which makes Wα as efficient as possible. Explain why the resulting formula makes sense.
Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. The observations are independent because they were randomly drawn. Consider the following two point estimators of the population mean μ: 1 = 0.10 X1 + 0.40 X2 + 0.40 X3 + 0.10 X4 and 2 = 0.20 X1 + 0.30 X2 + 0.30 X3 + 0.20 X4 Which of the following statements is true? HINT: Use the definition of...
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale. 3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
Let X1,, Xn be independent and identically distributed random variables with unknown mean μ and unknown variance σ2. It is given that the sample variance is an unbiased estimator of ơ2 Suggest why the estimator Xf -S2 might be proposed for estimating 2, justify your answer
Let X be a random variable with cdf FX (x:0), expected value EIX-μ and variance VlX- σ2. Let X1,X2, , Xn be an id sample drawn according to FX(x,8) where Fx (x,8) =万 for all x E (0,0). Let max(X1, X2, , X.) be an estimator of θ, suggested from pure common sense. Remember that if Y = max(X1, X2, , Xn). Then it can be shown that the cdf Fy () of Y is given by Fr(u) (Fx()" where...
Please solve these questions 1. Suppose that X1, X2, and Xs are random variables with common mean μ and variance matrix Find E(X1 +2X1X2-4X2X3 + X ]. 2. If X1, X2,..., X, are independent random variables with common mean (n - 1)] is an μ and variances σ?, σ2, .. ., σ unbiased estimate of varf , prove that Σ,(X,-X)2/[n 3. Suppose that in Exercise 2 the variances are known. Let X,-Σ,wa, be an unbiased estimate of μ (i.e., Σί...
If a null hypothesis is rejected at a significance level of 1%, then we should say that it was rejected at 1%. Reporting that the null was also rejected at the 5% level of significance is unnecessary and unwise. True False The p-value equals alpha, the level of significance of the hypothesis test. True False THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let X1, X2, X3, and X4 be a random sample of observations from a population with...
Let X1,X2,...,Xn denote independent and identically distributed random variables with mean µ and variance 2. State whether each of the following statements are true or false, fully justifying your answer. (a) T =(n/n-1)X is a consistent estimator of µ. (b) T = is a consistent estimator of µ (assuming n7). (c) T = is an unbiased estimator of µ. (d) T = X1X2 is an unbiased estimator of µ^2. We were unable to transcribe this imageWe were unable to transcribe...