Let X be a random variable and its probability density function (pdf) is
f(x) = θxθ−1 ; 0 < x < 1.
(a) Find the cumulative distribution function (cdf) of X, defined as FX(x) = P(X ≤ x).
(b) Define U = − ln(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 X1, X2, . . . , Xn be independent and identically distributed random variables from the distribution of X (not U). Answer the following questions.
i. Write down the log-likelihood function from X1, . . . , Xn.
ii. Hence find the maximum likelihood estimator for θ.
iii. Demonstrate that T =\sumn(upper limit), i=1 (lower limit) ln(Xi) is a sufficient statistic.
iv. Compute the Cram´er-Row lower bound for unbiased estimators of θ.
v. Find the asymptotic normal distribution for the maximum likelihood estimator of θ.
vi. Write down the formula for the 95% asymptotic confidence interval for θ.
(d) Assuming an estimate of θ is θb = 2, find an approximation to the probability P(X1 × X2 × · · · × X30 ≤ 1.85 × 10−5 ).
Express your answer using the N(0, 1) cdf Φ(x).