Question: Fill in the gaps in Example 2.3.10. (a) Show that if X1~ f1(x), then EXrl = er2/2, r = 0,1,.... So f1 (x) has all

Fill in the gaps in Example 2.3.10.
(a) Show that if X1~ f1(x), then
EXrl = er2/2, r = 0,1,....
So f1 (x) has all of its moments, and all of the moments are finite.
(b) Now show that

for all positive integers r, so EXr1 = EXr2 for all r. (Romano and Siegel 1986 discuss an extreme version of this example, where an entire class of distinct pdfs have the same moments. Also, Berg 1988 has shown that this moment behavior can arise with simpler transforms of the normal distribution such as X3.)

z"fi (z) sin(2 log x) dz = 0 J0

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