Homework 1 . Due Sept. 5, 2016 



1. If U is a uniform(0, 1) random variable, verify that log(U) is distributed as exp(1)
   [the natural log].
  
   Given any r.v. X, that is positive and continuous; find a transformation g( ) such that
   g(X) is distributed as exp(1).

 

2. If X_1, ... X_n are iid random variables with exp(lambda) distribution, please find

   E( n/sum X_i ) and Var( n/sum X_i )


3. (cont.) Verify the mle of lambda is n/sum X_i.
     Compute the approx. variance of mle by the inverse of the observed Fisher information.




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 Using the notations from my notes "Likelihood for Censored Data" on the relation between the distributions
 of (T, \delta) and (X, C) under random right censorship model, derive a formula that connecting 
 (that is, some function of) F_x( )  to the U_1( ) and U_0( ). i.e. F_x as a function of U_1, and U_0. 
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