Suppose that X',.X% are independent, both distributed normally with an unknown mean u and variance 4. a. Check ifXi +X2 is sufficient for μ. b. Give an unbiased estimator of u10. c. Is your e...
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...
1. (40) Suppose that X1, X2, .. , Xn, forms an normal distribution with mean /u and variance o2, both unknown: independent and identically distributed sample from 2. 1 f(ru,02) x < 00, -00 < u < 00, o20 - 00 27TO2 (a) Derive the sample variance, S2, for this random sample (b) Derive the maximum likelihood estimator (MLE) of u and o2, denoted fi and o2, respectively (c) Find the MLE of 2 (d) Derive the method of moment...
Suppose that X and Y are independent random variables with the same unknown mean u. Both X and Y have a variance of 36. Let T = aX + bY be an estimator of u. What condition must a and b satisfy in order that T be an unbiased estimator for ? Is T a normal random variable?
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
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?
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 X1 and X2 be independent random variables with mean μ and variance σ2. Suppose we have two estimators 1 (1) Are both estimators unbiased estimatros for θ? (2) Which is a better estimator?
x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate...
Suppose you have a sample of n independent observations X1,X2,...,Xn from a normal population with mean μ (known) and variance σ2 (unknown). (a) Find the ML estimator of σ2 . (b) Show that the ML estimator in (a) is a consistent estimator of θ. (c) Find a sufficient statistic for σ2. (d) Give a MVUE for θ based on the sufficient statistic.
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...