Page 26 - Elements of Distribution Theory
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12 Properties of Probability Distributions
a countable collection. Although this does hold, the proof is more complicated and it is
useful, if not essential, to use a more sophisticated method of proof.
A number of useful properties of distribution functions follow from the fact a distribution
function is nondecreasing. The following result gives one of these.
Theorem 1.5. Let F denote the distribution function of a distribution on R. Then the set of
points x at which F is not continuous is countable.
Proof. Let D denote the set of points at which F has a discontinuity. For each positive
integer m, let D m denote the set of points x in R such that F has a jump of at least 1/m at
x and let n m denote the number of elements in D m . Note that
∞
D = D m
m=1
since
lim F(x) = 1 and lim F(x) = 0,
x→∞ x→−∞
n m ≤ m.It follows that the number of points of discontinuity is bounded by ∞ m. The
m=1
result follows.
Discrete distributions
Hence, although a distribution function is not necessarily continuous, the number of jumps
must be countable; in many cases it is finite, or even 0. Let X denote a real-valued random
variable with distribution function F.If F is a step function, we say that the X has a discrete
distribution or is a discrete random variable.
Example 1.11 (Integer random variable). Let X denote a random variable with range
X ={1, 2,..., m} for some m = 1, 2,..., and let
θ j = Pr(X = j), j = 1,..., m.
The distribution function of X is given by
0 if x < 1
if 1 ≤ x < 2
θ 1
θ 1 + θ 2
if 2 ≤ x < 3
F(x) = .
.
.
θ 1 +· · · + θ m−1
if m − 1 ≤ x < m
1 if m ≤ x
where θ 1 ,...,θ m are constants summing to 1. Hence, F is a step function and X has a
discrete distribution.
Distribution functions for random vectors
d
Fora random vector X taking values in R , the distribution function is defined as the
d
function F : R → [0, 1] given by
F(x) = Pr{X ∈ (−∞, x 1 ] × (−∞, x 2 ] × ··· (−∞, x d ]}, x = (x 1 ,..., x d ).
