(d) Is it easy to study the finite-sample distributional property of the MLE Â, such as E(1), var(Â), the mean square error (MSE) in estimating À, and its exact sampling distribution?
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- Let X₁, X2,..., Xñ be a random sample from the exponential distribution with rate parameter A. (a) Find the CRLB for the variance of all unbiased estimators of X. (b) Find the efficiency of the MLE of X. (c) Find the CRLB for the variance of all unbiased estimators of E(X) = 1/λ. (d) Can you find an estimator of E(X) = 1/λ that is efficient? Justify your answer.7. Let X be a continuous random variable with pdf f(x)=√ 0Derive a general expression for the variance of net future loss Var(L) with a constant force of interest δ under each of the following models:i. μ_x(t) = μii. μ_x(t) = 1/(ω-x)Let Y be a random variable with pdf f(y)= (1/18)(y^3−1) for 1<y<3.(a) Calculate P (Y > 2.5).(b) Calculate the variance of Y .The pdf is given as follows: f(x) = { (1/6)e^(1/6x) ; x > 0 0; ew] (i) Find the mean and variance of X using the pdf. (ii) Find the moment generating function of X. (iii) Find the mean and variance of X using the moment generating function.A time series dataset consists of 100 observations x, x,., j09. The following summary statistics have been made derived for this data: 100 100 100 E*, = 1,000; (x, - x) = 280;(x, -F)(x-X) = 50:E(-(*-)=-250. a) Calculate the sample autocorrelation functions at lags 1 and 2.The random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−3/4. Calculate Var(X+Y) with a full explanationThe p.d.f. of a random variable X' is as shown in the figure. The pdf is zero for X 5. Calculate (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X. fx (x) kt x, x, ... x be a random sample from an exponential distribution with parameter 0. Find sufficient estimator for '8'.SEE MORE QUESTIONS