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374 Appendix A Random Variables and Probability Distributions
Example A.1.2 The Poisson distribution
A.2 Random Vectors
The mean of the Poisson distribution with parameter λ (see Example (h) above) is
given by
µ
∞
j0
jλ j
j! e−λ
∞
j1
λλ j−1
(j − 1)! e−λ λe λ e −λ λ.
A similar calculation shows that the variance is also equal to λ (see Problem A.2).
Remark. Functions and parameters associated with a random variable X will be
labeled with the subscript X whenever it is necessary to identify the particular random
variable to which they refer. For example, the distribution function, pdf, mean, and
variance of X will be written as FX, fX, µX, and σ 2 X
, respectively, whenever it is
necessary to distinguish them from the corresponding quantities FY , fY , µY , and σ 2 Y
associated with a different random variable Y .
An n-dimensional random vector is a column vector X (X1,...,Xn) ′ each of whose
components is a random variable. The distribution function F of X, also called the
joint distribution of X1,...,Xn, is defined by
F(x1,...,xn) P [X1, ≤ x1,...,Xn ≤ xn] (A.2.1)
for all real numbers x1,...,xn. This can be expressed in a more compact form as
F(x) P [X ≤ x], x (x1,...,xn) ′ ,
for all real vectors x (x1,...,xn) ′ . The joint distribution of any subcollection
Xi1 ,...,Xik of these random variables can be obtained from F by setting xj ∞in
(A.2.1) for all j/∈{i1,...,ik}. In particular, the distributions of X1 and (X1,Xn) ′ are
given by
and
FX1(x1) P [X1 ≤ x1] F(x1, ∞,...,∞)
FX1,Xn(x1,xn) P [X1 ≤ x1,Xn ≤ xn] F(x1, ∞,...,∞,xn).
As in the univariate case, a random vector with distribution function F is said to be
continuous if F has a density function, i.e., if
F(x1,...,xn)
xn
−∞
···
x2 x1
−∞
f(y1,...,yn)dy1dy2 ···dyn.
−∞