Full solve good handwritting

Problem 2. (12 points) Let X1, X2, . ... Xy be a random sample of size 7 taken from a population, which follows the distribution 9 (z; t) = if x > 0, 0 otherwise. It is given that the mean and standard deviation of this population are equal to unknown parameter t, where t > 0. a. Which one of the following estimator of 2t# is unbiased? O X3 - X5 O X3 + Xs O X3 0 7.X b. Compare the efficiencies of the unbiased estimators X, and -5.X2 + 6.X3 of the population mean t. The unbiased estimator X of the population meant ? the unbiased estimator -5.X2 + 6.X, of the population mean t. c. It is given that the samples values are as follows: X1 = 8.04, X2 = 2.07, X3 - 5.54, X4 - 7.8, X's = 6.18, X6 - 5.39, X7 = 1.43. (1) Find the likelihood function L (t). L (t) - (il) Find the log-likelihood function I (t). 1 (t) = (ii) Find the derivative I'(t) of the log-likelihood function (with respect to t). I'(t) = (iv) Find the maximum likelihood estimate t of t. d. Compare the widths of the 68.01% and 67.62% confidence intervals for the parameter t. The width of the 68.01% confidence interval for t is |? the width of the 67.62% confidence interval for t. e. If (0, 9.77) is a 98.5% confidence interval for t, then find the following probability: P(t > 9.77) =