consider again a random sample X₁,..., X, from the exponential distribution with density and parameter à Є (0,00) f(x; 0) = e, x € (0,00), (1) Show that X, is a complete statistic for A. (2) Find an UMVUE for A. (3) Calculate E(nX(1)|X) explicitly
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- 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 X1, X2,..., X, be a random sample from a distribution with a pdf ke-kx x > 0 f (x) = 0, elsewhere. (a) Is ô = E-; X; an unbiased estimator of ? Justify. i=i (b) Is â = min{X1, X2,... , Xn} an unbiased estimator of t? Justify.Two real-valued RVs, X and Y, have joint PDF 1 p(x1, x2) = exp 2TV1- 2(1 - r?) where -1Let 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.Let 0, X,, X2,... be RVs. Suppose that, conditional on = 0, X1, X2, ... are independent and X, is normally distributed with mean 0 and variance o. Suppose that the marginal PDF of O is 1 T(0) = e 0ER. Calculate the mean and variance of O conditional on X, =x, ..., X,= x„.Show that the mean x of a sample of size n from the distribution. e-x/0 f (x)=- A i 0Consider a set of data x1, x2, n n i=1 ..., n taken from a population with mean µ. - Show that (x-μ)² = Σ(x₂ − x)² + n(x − µ)². i=1Prob. 3 Let X be a random variable with cumulative distribution function (cdf) given by (1-e-x², x ≥ 0 ={1,- x<0 Find the probability that the random variable X falls within one standard deviation of its mean. Fx (x) =Consider a random sample from NB(r, p) where the parameter r is known to be 3. An experiment is run with n = 10 trials and the sample mean is observed to be x̄ = 0.6. (a) Derive a formula for the MLE p̂ as a function of n, r and X̄ . (b) Find the estimate of p. (c) Find the MLE for the population mean.SEE MORE QUESTIONSRecommended textbooks for youGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill