Here has mean and variance .
Now the sample mean is
i) The expected value of is
The variance of is
\
ii) Consider the estimator is
The expected value of is
Hence is an unbiased estimator of .
The variance is
Since the variance of is lesser than that of
we prefer the estimator .
Let Y, Y2, Yz and Y4 be independent, identically distributed random variables from a population with...
Let Yı, Y2, Ys, and Y4 be independent, identically distributed random variables from a mean u and a variance 02. Consider a different estimator of u: W=Y+Y2+2Y3+ Y 00 This is an example of a weighted average of the Y a) Show that W is a linear estimator. b) Is W an unbiased estimator of u? Show that it is - or it isn't (E(W) = Find the variance of W and compare it to the variance of the sample...
Let Yi, Y2,.... Yn denote independent and identically distributed uniform random variables on the interval (0,4A) obtain a method of moments estimator for λ, λ. Calculate the mean squared error of this estimator when estimating λ. (Your answer will be a function of the sample size n and λ
Let Yi, Y2,.... Yn denote independent and identically distributed uniform random variables on the interval (0,4A) obtain a method of moments estimator for λ, λ. Calculate the mean squared error of this estimator when estimating λ. (Your answer will be a function of the sample size n and λ
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
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
Question 3 15 marks] Let X1,..,X be independent identically distributed random variables with pdf common ) = { (#)%2-1/64 0 fx (a;e) 0 where 0 >0 is an unknown parameter X-1. Show that Y ~ T (}, ); (a) Let Y (b) Show that 1 T n =1 is an unbiased estimator of 0-1 ewhere / (0; X) is the log- likeliho od function; (c) Compute U - (d) What functions T (0) have unbiased estimators that attain the relevant...
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...
9. Let X and Y be independent and identically distributed random variables with mean u and variance o. Find the following: (a) E[(x + 2)] (b) Var(3x + 4) (c) E[(X-Y)] (d) Cov{(X + Y), (X - Y)}
Suppose Y1, Y2, …, Yn are independent and identically distributed random variables from a uniform distribution on [0,k]. a. Determine the density of Y(n) = max(Y1, Y2, …, Yn). b. Compute the bias of the estimator k = Y(n) for estimating k.
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...