Use the result of Exercise 2.5-5 to find the mean and variance of the
(a) Bernoulli distribution
(b) Binomial distribution.
(c) Geometric distribution.
(d) Negative binomial distribution.
Reference Exercise 2.5-5:
Let the moment-generating function M(t) of X exist for -h < t < h. Consider the function R(t) = M(t). The first two derivatives of R(t) are, respectively:
R'(t) = M'(t)
R''(t) = M(t)M''(t) + [M'(t)]^2
and
Setting t = 0, show that
(a) p = R'(0).
(b) σ^2 = R''(0).