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

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Let X be a random variable and its probability density function (pdf) is f(x) = 0x¹; 0<x< 1. (a) Find the cumulative distribution function (cdf) of X, defined as Fx(x) = P(X ≤ 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 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 0. iii. Demonstrate that T = ln(X) is a sufficient statistic. iv. Compute the Cramér-Row minimum variance lower bound for unbiased esti- mators of 0. v. Find the asymptotic normal distribution for the maximum likelihood estima- tor of 0. vi. Write down the formula for the 95% asymptotic confidence interval for 0. Let X be a random variable and its probability density function (pdf) is f(x) = 0x¹; 0<x< 1. (a) Find the cumulative distribution function (cdf) of X, defined as Fx(x) = P(X ≤ 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 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 0. iii. Demonstrate that T = ln(X) is a sufficient statistic. iv. Compute the Cramér-Row minimum variance lower bound for unbiased esti- mators of 0. v. Find the asymptotic normal distribution for the maximum likelihood estima- tor of 0. vi. Write down the formula for the 95% asymptotic confidence interval for 0.

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a The cdf of X is FXx 0x fx dx 0x x1 dx 1 x b The cdf of U lnX ... View the full answer