I really do not understand how to prove Rt mean nor how Rt s

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

Solution