(c) Let 7 denote the population median. Obtain the MLE of T. (Hint: First find the specific form of the median in terms of A and utilize the invariance property of MLE) (d) Is it easy to study the finite-sample distributional property of the MLE Â. such as E(X).
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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)=√ 0Let 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.Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.Derive 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)Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.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 explanationSEE MORE QUESTIONS