Page 38 - Elements of Distribution Theory

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24 Properties of Probability Distributions

one might conclude that the distributions are very similar; however, the density functions
indicate important differences in the distributions.

In giving expressions for density functions, it is often convenient to give the value of the
density function, p(x), only for those values of x for which the value is nonzero. For instance,
in the previous example, the density function might be given as p(x) = 6(2 − x)(x − 1),
1 < x < 2; this statement implies that for x ≤ 1or x ≥ 2, p(x) = 0.

Discrete distributions
A second important special case occurs when F is a step function. Suppose that F has
jumps at x 1 , x 2 ,.... In this case,

F(x) = p(x j )
j:x j ≤x
where p(x j )is the size of the jump of F at x j . Hence,

p(x j ) = Pr(X = x j ), j = 1,....
In this case, X is a discrete random variable and the function p will be called the frequency
function of the distribution. The set of possible values of X is given by X ={x 1 , x 2 ,...,}.

Example1.23 (Binomialdistribution). Let X denotetherandomvariabledefinedinExam-
ple 1.4. Then X ={0, 1,..., n} and
n x n−x

Pr(X = x) = θ (1 − θ) , x = 0, 1,..., n;
x
here 0 <θ < 1isa constant. Hence, the distribution function of X is given by

n j n− j
F(x) = θ (1 − θ)
j
j=0,1,...,n; j≤x
so that X is a discrete random variable with frequency function

n x n−x
θ (1 − θ) , x = 0, 1,..., n.
x
A random variable can be neither discrete nor absolutely continuous.

Example 1.24 (A distribution function with discrete and absolutely continuous compo-
nents). Let X denote a real-valued random variable such that, for any A ⊂ R,
1 1
Pr(X ∈ A) = I {0∈A} + exp(−t) dt.
2 2 A∩(0,∞)
Thus, X is equal to 0 with probability 1/2; if X is not equal to 0 then X is distributed like
a random variable with probability function

exp(−t) dt.
A∩(0,∞)