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Unformatted text preview: SOLUTIONS FOR PROBLEM SET 1 STAT 5102-004 Given a parametric family of distributions, for parameter value and observable value y , let lik( | y ) = f ( y | ) be the likelihood at given y . Varying the parameter describes the likelihood function. (1) For any positive , a continuous uniform random variable Y over (0 , ) has density f ( y | ) = 1 I (0 , ) ( y ) in the real line. The indicator maps to one if the inequalities 0 < y < < are all satisfied and zero otherwise. Thus, for any positive y and , I (0 , ) ( y ) = I ( y, ) ( ) and lik( | y ) = 1 I ( y, ) ( ) . The following figure graphs the function. lik ( |y 29 y 2y 3y 4y 1 y (2) Densities of distributions in an exponential family have the form h ( y )exp ( y- ( ) ) . Parametric Free Canonical Cumulant Family Parameter Parameter Function Binomial( n,p ) p (0 , 1) = logit p ( ) = n log(1 + e ) Poisson( ) (0 , ) = log (...
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- Spring '03