Page 27 - Elements of Distribution Theory

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052184472Xc01 CUNY148/Severini May 24, 2005 17:52

1.4 Distribution Functions 13

If X is written in terms of component random variables X 1 ,..., X d each of which is real-
valued, X = (X 1 ,..., X d ), then
F(x) = Pr(X 1 ≤ x 1 ,..., X d ≤ x d ).

Example 1.12 (Two-dimensional polytomous random vector). Consider a two-
dimensional random vector X with range

2
X ={x ∈ R : x = (i, j), i = 1,..., m; j = 1,..., m}
and let

θ ij = Pr{X = (i, j)}.

The distribution function of X is given by
0 if x 1 < 1or x 2 < 1



 θ 11 if 1 ≤ x 1 < 2 and 1 ≤ x 2 < 2




 θ 11 + θ 12 if 1 ≤ x 1 < 2 and 2 ≤ x 2 < 3

 .

 .
 .




θ 11 +· · · + θ 1m if 1 ≤ x 1 < 2 and m ≤ x 2
F(x) = . , x = (x 1 , x 2 ).
 .
 .


 θ 11 +· · · + θ m1 if m ≤ x 1 and 1 ≤ x 2 < 2



 θ 11 +· · · + θ m1 + θ 12 +· · · + θ m2
 if m ≤ x 1 and 2 ≤ x 2 < 3

 .

 .
 .


1 if m ≤ x 1 and m ≤ x 2
This is a two-dimensional step function.
Example 1.13 (Uniform distribution on the unit cube). Consider the random vector X
3
defined in Example 1.5. Recall that X has range X = (0, 1) and for any subset A ⊂ X,

Pr(X ∈ A) = dt 1 dt 2 dt 3 .
A
Then X has distribution function

x 3 x 2 x 1
F(x) = dt 1 dt 2 dt 3 = x 1 x 2 x 3 , x = (x 1 , x 2 , x 3 ), 0 ≤ x j ≤ 1, j = 1, 2, 3
0 0 0
with F(x) = 0if min(x 1 , x 2 , x 3 ) < 0. If x j > 1 for some j = 1, 2, 3, then F(x) =
x 1 x 2 x 3 /x j ;if x i > 1, x j > 1 for some i, j = 1, 2, 3, then F(x) = x 1 x 2 x 3 /(x i x j ).
d
Like distribution functions on R,a distribution function on R is nondecreasing and
right-continuous.
Theorem 1.6. Let F denote the distribution function of a vector-valued random variable
d
X taking values in R .
d
(i) If x = (x 1 ,..., x d ) and y = (y 1 ,..., y d ) are elements of R such that x j ≤ y j ,
j = 1,..., d, then F(x) ≤ F(y).