Question

he second form for one-parameter exponential family distributions, introduced during lecture 09.1, was Jy (y | θ) = b(y)ec(0)t(y)-d(0) Let η = c(0). If c is an invertible function, we can rewrite (1) as where η is called the natural, or canonical, parameter and K(n) = d(C-1(n)). Expression (2) is referred to as the canonical representation of the exponential family distribution (a) Function κ(η) is called the log-normalizer: it ensures that the distribution fy(y n) integrates to one. Show that, for continuous random variables Note: For discrete random variables, the integral is replaced by a summation (b) It can be shown that in canonical exponential family distributions, the function K is a generating function for the moments of t(Y). Show that for continuous random variables d2n(n) = Var[t(Y)] d)and f*( c) These equalities also hold for discrete random variables. We have shown in class that the binomial distribution is a one-parameter exponential family distribution, since it can be written as Derive the corresponding canonical representation. Clearly specify vour function η c(p), and the functions b(y), t(y) and K(η). Simplify n(η) as much possible Determine Elt(Y] and Var!t(Y)] as a function of η using equations (3) . Now determine E(Y) and Var[t(Y)| as a function of p, using the substitution η c(p). What do you find? Is this an expected finding?

Answer 1

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ML lteM l+e Ite) t e l+en lteM te