Applied Statistics Using SPSS, STATISTICA, MATLAB and R

422 Appendix A - Short Survey on Probability Theory
[−1.96σ, 1.96σ] (see normal distribution tables), we then have: ε = 1.96σ = 0.0196
(smaller than the previous “model-free” estimate).
a
0.5
0.4
0.3
0.2
0.1
0
f (x )
-3 -2 -1 0 1 2 x 3
b
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
F (x )
-3 -2 -1 0 1 2 x 3
Figure A.7. The standard normal density (a) and distribution (b) functions.
Example A. 19
Q: Let X be a standard normal variable. Determine the density of Y = X 2 and its
expectation.
A: Using the previous result of Example A.11:
1 − y / 2
[ f ( y ) + f ( − y ) ] = e y > 0
1
g(
y)
=
2 y
2πy
.
This is the density function of the so-called chi-square distribution with one
degree of freedom.
Ε Y =
∞
yg(
y)
dy = 1/
2π
∞ − y / 2
y e dy . Substituting y
∫ ∫
The expectation is: [ ] ( )
by x 2 , it can be shown to be 1.
A.8 Multivariate Distributions
A.8.1 Definitions
0 0
A sequence of random variables X1, X2,…, Xd, can be viewed as a vector
x = [ X 1,
X 2 , K X d ] with d components. The multivariate (or joint) distribution
function is defined as:
F x , x , K x ) = P(
X ≤ x , X ≤ x , K,
X ≤ x ) . A. 41
( 1 2 d 1 1 2 2 d d