I really do not understand how to prove R\'(t)= mean nor how R\'\'(t)= std devat

Question

I really do not understand how to prove R'(t)= mean nor how R''(t)= std devation?

Let the moment-generating function M(t) of X exist for -h <t < h. Consider the function R(t) = ln M(t). The first twoderivatives of R(t) are respectively,

R'(t) =M'(t)    and     R"(t) =M(t)M"(t) - [M'(t)]2

            M(t)                                    [M(t)]2

1. Setting t = 0, show that

a). = R'(0)

b). 2 = R"(0)

2. Use the results from above to find the mean and variance ofthe

a). Bernoulli distribution

b). Binomial distribution

c). Geomentic distribution

d). Negative binomial distribution

Explanation / Answer

I really do not understand how to prove R'(t)= mean nor how R''(t)= std devat

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