Page 32 - Elements of Distribution Theory

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

18 Properties of Probability Distributions

random variable with

n x n−x
Pr(X = x) = θ (1 − θ) , x = 0,..., n
x
where 0 <θ < 1. Let m 0 denote the largest positive integer for which

m 0
n j n− j 1
θ (1 − θ) ≤ .
j 2
j=0
If
m 0
n j n− j 1
θ (1 − θ) < ,
j 2
j=0
then the median of the distribution is m 0 + 1; otherwise, any value in the interval (m 0 ,
m 0 + 1) is a median of the distribution.

There are a number of properties that any quantile function must satisfy; for convenience,
we use the convention that Q(0) =−∞.

Theorem 1.8. Consider a real-valued random variable X with distribution function F and
quantile function Q. Then
(i) Q(F(x)) ≤ x, −∞ < x < ∞
(ii) F(Q(t)) ≥ t, 0 < t < 1
(iii) Q(t) ≤ xif and only if F(x) ≥ t
(iv) If F −1 exists, then Q(t) = F −1 (t)
(v) If t 1 < t 2 , then Q(t 1 ) ≤ Q(t 2 )

Proof. Define the set A(t)by

A(t) ={z: F(z) ≥ t}

so that Q(t) = inf A(t). Then A[F(x)] clearly contains x so that Q[F(x)] = inf A[F(x)]
must be no greater than x; this proves part (i). Note that if F(x) = 0, then A(t) = (−∞, x 1 ]
for some x 1 so that the result continues to hold if Q(F(x)) is taken to be −∞ in this case.
Also, for any element x ∈ A(t), F(x) ≥ t; clearly, this relation must hold for any
sequence in A(t) and, hence, must hold for the inf of the set, proving part (ii).
Suppose that, for a given x and t, F(x) ≥ t. Then A(t) contains x; hence, Q(t) ≤ x.
Now suppose that Q(t) ≤ x; since F is nondecreasing, F(Q(t)) ≤ F(x). By part (ii) of the
theorem F(Q(t)) ≥ t so that F(x) ≥ t, proving part (iii).
If F is invertible, then A t ={x: x ≥ F −1 (t)} so that Q(t) = F −1 (t), establishing
part (iv).
Let t 1 ≤ t 2 . Then A(t 1 ) ⊃ A(t 2 ). It follows that the

inf A(t 1 ) ≤ inf A(t 2 );

part (v) follows.