Question: 1. Let fo be a probability density. An exponentially tilted or weighted density f based on fo and with parameter 0 is given by f(I;

1. Let fo be a probability density. An exponentially tilted or

1. Let fo be a probability density. An exponentially tilted or weighted density f based on fo and with parameter 0 is given by f(I; 0) = exp(@:) fo(I) M(0) where M(@) is a normalization factor that guarantees ff(; 0)dr = 1. (a) Show that M is the moment generating function of fo. (b) Show that f belongs to an exponential family with a() = 1. What is the cumulant function b(0) of f? (c) Now do GLM' problem 2.14. (Hint: to compute M(@) use the substitution u = ex/(1 + er) and note that the resulting integrand is related to the beta distribution.)

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