Applied Statistics Using SPSS, STATISTICA, MATLAB and R

n [ X ] = ∑ i=
1
A.6 Expectation, Variance and Moments 417
v ≡ V ( x − x)
f
ˆ . A. 28
i
2
i
This is the so-called sample variance. The square root of v, s = v , is the
sample standard deviation. In Appendix C we present a better estimate of v.
The variance can be computed using the second order moment, observing that:
2 2
2 2 2
[ ] = Ε[
( X − µ ) ] = Ε[
X ] − 2µ
Ε[
X ] + µ = Ε[
X ] −
V X µ . A. 29
4. Gauss’ approximation formulae:
i. [ g( X ) ] ≈ g(
Ε[
X ])
Ε ;
2
⎡dg
⎤
g ⎢ ⎥ .
⎢⎣
dx Ε[ X ] ⎥⎦
ii. V[
( X ) ] ≈ V[
X ]
A.6.2 Moment-Generating Function
The moment-generating function of a random variable X, is defined as the
tX
expectation of e (when it exists), i.e.:
tX
ψ ( t) = Ε[
e ] . A. 30
X
The importance of this function stems from the fact that one can derive all
moments from it, using the result:
k [ X ]
n
d ψ X ( t)
Ε =
. A. 31
n
dt
t=
0
A distribution function is uniquely determined by its moments as well as by its
moment-generating function.
Example A. 15
Q: Consider a random variable with the Poisson probability function
−λ
k
P(
X = k)
= e λ / k!
, k ≥ 0. Determine its mean and variance using the momentgenerating
function approach.
A: The moment-generating function is:
tX ∞ tk −λ
k −λ
[ e ] = ∑ e e λ / k!
= e ∑
k = 0
∞
k = 0
ψ ( t) = Ε
( λe
)
X
t
k
/ k!
.
Since the power series expansion of the exponential is ∑ ∞ x
k
e = x / k!
one
k = 0
can write:
−λ
λe
t
λ(
e
t
−1)
ψ ( t)
= e e = e .
X