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Unformatted text preview: ACTSC 432 - Loss Models 2 TEST #1 Name : ID Number : 1. (5 marks) (a) Determine whether the binomial distribution with probability mass function f ( x ; & ) = & m x ¡ & x (1 & & ) m & x , x = 0 ; 1 ;:::;m , (1) for < & < 1 and m 2 f 1 ; 2 ;::: g is a member of the linear exponential family. If it is, identify the functions p ( x ) , q ( & ) , and r ( & ) of the linear exponential family representation. Solution: The binomial p.m.f. can be expressed as f ( x ; & ) = & m x ¡ & x (1 & & ) m & x = & m x ¡ (1 & & ) m & & 1 & & ¡ x = ¢ m x £¢ & 1 & & £ x (1 & & ) & m = ¢ m x £ e x ln ( & 1 & & ) (1 & & ) & m . Thus, f ( x ; & ) = p ( x ) e r ( & ) x q ( & ) , where p ( x ) = & m x ¡ , q ( & ) = (1 & & ) & m , and r ( & ) = ln & & 1 & & ¡ = ln & & ln (1 & & ) . Indeed, the binomial distribution is a member of the linear exponential family. 1 (5 marks) (b) Assume that X j & = & has probability mass function f ( x ; & ) given in (1) and & has a beta distribution with mean ¡= ( ¡ +...
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- Spring '09
- Probability theory, Probability mass function, Poisson random variables, linear exponential family, claims basis