Applied Statistics Using SPSS, STATISTICA, MATLAB and R

Example A. 12
A.6 Expectation, Variance and Moments 415
Q: A gambler throws a dice and wins 1€ if the face is odd, loses 2.5€ if the face is
2 or 4, and wins 3€ if the face is 6. What is the gambler’s
expectation?
A: We have:
⎧ 1 if X = 1,
3,
5;
⎪
g( x ) = ⎨−
2.
5 if X = 2,
4;
⎪
⎩ 3 if X = 6.
1 2.
5 3 1
Therefore: Ε[ g ( X ) ] = 3 − 2 + = .
6 6 6 6
The word “expectation” is somewhat misleading since the gambler will only
expect to get close to winning 1/6 € in a long run of throws.
The following cases are worth noting:
1. g(X) = X: Expected value, mean or average of X.
µ = Ε[
X ] = ∑ x P(
X = x ) , if X is discrete (and the sum exists); A.24a
[ ] ∫ ∞
=
−∞
i
i
i
µ = Ε X xf ( x)
dx , if X is continuous (and the integral exists). A.24b
The mean of a distribution is the probabilistic mass center (center of gravity) of
the distribution.
Example A. 13
Q: Consider the Cauchy distribution, with:
mean?
A: We have:
f X
a
( x)
=
π a
2
1
+ x
2
1 2
, x∈ℜ. What is its
[ ] ∫ ∞ a x
Ε X =
π − ∞ 2 2
a + x
dx.
But
x
∫ 2 2
a + x
2
dx = ln( a + x
2
) , therefore the
integral diverges and the mean does not exist.
Properties of the mean (for arbitrary real constants a, b):
i. [ aX + b]
= aΕ[
X ] + b
ii. [ X + Y ] = Ε[
X ] + Ε[
Y ]
iii. [ XY ] = Ε[
X ] Ε[
Y ]
Ε (linearity);
Ε (additivity);
Ε if X and Y are independent.
The mean reflects the “central tendency” of a distribution. For a data set with n
values xi occurring with frequencies fi, the mean is estimated as (see A.24a):