3. Consider the natural form of the one parameter exponential family (1PEF): f(x|n) = h(x)c* (n)exp[nT(x)], xe R,NEH, wh

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3 Consider The Natural Form Of The One Parameter Exponential Family 1pef F X N H X C N Exp Nt X Xe R Neh Wh 1 (49.94 KiB) Viewed 91 times

3. Consider the natural form of the one parameter exponential family (1PEF): f(x|n) = h(x)c* (n)exp[nT(x)], xe R,NEH, where (c* (n)]-1 = h(x))exp[nT(x)]dx, and, H is the natural parameter space. Let A(n) = log(c* (n)). Express f (x|n) using An). a. If n is an interior point of natural parameter space (H), then find the moment generating function of T(X), where X ~ f(x0). b. Using 3(a), show that E(T(X)] = A' (n), where ' denotes derivative. c. Using 3(a), show that var[T(X)] = A" (n). d. Apply 3(b) and 3(C) on the pdf given in exercise 1(d).