Let X be a random variable and its probability density function (pdf) is f(x) = 0r#-1; 0

Let X be a random variable and its probability density function (pdf) (a) Find the cumulative distribution function (cdf) of X, defined as = P(X (b) Define U = In(X). First find the cdf of U and then find the pdf of U. What is the distribution of U? (Need to specify both distribution name and parameter(s).) (c) Let Xl,X2,.. , Xn be independent and identically distributed random variables from the distribution of X (not U). Answer the following questions. Write down the log likelihood function from XL, , X Hence find the maximum likelihood estimator for g. n. Ili. Demonstrate that T = In(Xi) is a sufficient statistic. Compute the Cramr-Row lower bound for unbiased estimators of 9. Find the asymptotic normal distribution for the maximum likelihood estima- v. tor of 9. vi. Write down the formula for the 95% asymptotic confidence interval for 9. (d) Assuming an estimate of is i = 2, find an approximation to the probability P(XI x & x 1.85 x 10-5). Express your answer using the N (O, 1) cdf