Page 104 - Probability and Statistical Inference

2. Expectations of Functions of Random Variables 81

Observe that log(M (t)) = nlog{(1 – p) + pe } so that we obtain
t
X

Next, using the chain rule of differentiation, from (2.3.6) we can write

From (2.3.7) it is obvious that dM (t/dt, when evaluated at t = 0, reduces to
X

We found the mean of the distribution in (2.2.14). Here we have another
way to check that the mean is np.
Also, the Theorem 2.3.1 would equate the expression of E(X ) with d M (t)/
2
2
X
dt evaluated at t = 0. From (2.3.6) it is obvious that d M (t)/dt , when evalu-
2
2
2
X
ated at t = 0, should lead to η , that is
2

Hence, one has V(X) = E(X ) – µ = np – np + n p – (np) = np–np = np(1
2
2
2 2
2
2
2
– p). This matches with the expression of σ given in (2.2.17).
2
2.3.2 The Poisson Distribution
–λ
Suppose that X has the Poisson(λ) distribution with its pmf f(x) = e λ /x! for x
x
= 0, 1, ... and 0 < λ < ∞, given by (1.7.4). Here, for all fixed t ∈ ℜ we can
express M (t) as
X

t
Observe that log(M (t)) = –λ + λe so that one has
X