Distribution functions

14.5 Distribution functions

We turn now to the problem of computing the probabilities of events associated with

be a one-dimensional random variable defined on a sample space S, where S is a Bore1 set in n-space for some

a given random variable. Let

be a probability measure defined on the

. Let P

is a real number, and as

subsets of S. For each

in S,

runs through the elements of S the numbers run through a set of real numbers (the range of X). This set may be finite, countably infnite, or uncountable. For each real number we consider the following special subset of S:

= X(0) t).

If is less than all the numbers in the range of

will be empty; otherwise, will be a

the set

subset of S. We assume that for each the set

is an event,

515 that is, a

Distribution functions

set. According to the convention discussed at the end of Section 14.3, we denote this event by the symbol X Suppose we know the probability

for every real We shall find in a moment that this knowledge enables us to compute the probabilities of many other events of interest. This is done by using the probabilities

as a basis for constructing a new function F, called the distribution function of

It is defined as follows:

be a one-dimensional random variable. The function F

DEFINITION OF A DISTRIBUTION FUNCTION.

Let

for all real t by the equation

F(t) =

t)

is called the distribution function of the random variable Note: Sometimes the notation

is used to emphasize the fact that the distribution function is associated with the particular random variable

The value of the function at is then denoted by

It is important to realize that the distribution function F is defined over the entire real axis, even though the range of X may be only a bounded portion of the real axis. In fact, if all numbers

lie in some finite interval [a, b], then for t < a the probability t) is zero (since for t < a the set X

b the probability t) is (because in this case the set X

t is empty) and for t

t is the entire sample space). This means that for bounded random variables X whose range is within an interval [a, b] we have F(t) = 0 for all

We now proceed to derive a number of properties common to all distribution functions.

T H EO R EM

a one-dimensional random variable Then we have:

14.2. Let F denote a distribution function of

for all

(b) < = F(b) F(a) if

Proof. Part (a) follows at once from the definition of F because probabilities always lie between 0 and 1. To prove (b) we note that the events “a <

a” are disjoint. Their union is the event “X

and “X

b.” Using additivity we obtain

which can also be expressed as

Part (c) follows from (b) since

b)

Note: Using the mass analogy, we would say that F(t) represents the total amount of mass located between and (including the point itself). The amount of mass located in a half-open interval (a, b] is F(b) F(a).

Calculus of

1 ---------

14.2 A distribution function of an of a bounded random variable.

F IGURE 14.1 A distribution function

unbounded random variable.

Figure 14.1 shows a distribution function of a bounded random variable whose values

lie in the interval [0, This particular example is known as a uniform distribution. Here we have

for

F(t) = t for 0

F(t) = for t

Figure 14.2 shows an example of a distribution function corresponding to an unbounded random variable. This example is known as a Cauchy distribution and its function values are given by the formula

F(t) = +

Experiments that lead to uniform and to Cauchy distributions will be discussed later.

Note: Using the mass analogy, we would say that in Figure 14.1 no mass has been placed to the left of the origin or to the right of the point 1. The entire mass has been distributed over the interval [0,

The graph of is linear over this interval because the

mass is smeared with a uniform thickness. In Figure 14.2 the mass has been smeared along the entire axis. The graph is nonlinear because the mass has been smeared with a varying

thickness. Theorem 14.2(b) tells us how to compute (in terms of F) the probability that X lies in a

half-open interval of the form (a, b]. The next theorem deals with other types of intervals.

THEOREM 14.3. Let F be a distribution function of a one-dimensional random variable Then

b we have: (a) P(a

b) = F(b)

F(a) + P(X = a).

(b) P(a < < b) = F(b) F(a)

= b).

(c) P(a < b) = F(b) F(a) +

= a)

= b).

Proof. To prove (a) we note that the events “a < b” and “X = a” are disjoint and their union is

b Using additivity and Theorem 14.2(b) we obtain (a). Parts (b) and (c) are similarly proved.

517 Note that all four events

Discontinuities of distribution functions

and

have equal probabilities if and only if

= a) = 0 and

The examples shown in Figures 14.1 and 14.2 illustrate two further properties shared by all distribution functions. They are described in the following theorem.

a one-dimensional random variable Then we have

THEOREM

14.4. Let F be the distribution function of

Proof. The existence of the two limits in (14.1) and the fact that each of the two limits

lies between 0 and follow at once, since F is a monotonic function whose values lie between

0 and Let us denote the limits in (14.1) by

respectively. To prove that = 0 and that

and

= 1 we shall use the countably additive property of probability. For this purpose we express the whole space S as a countable union of disjoint events:

Then, using additivity, we get

The sums on the right will telescope, giving us P(S) = lim [F(O)

+ lim

F(O)]

Since P(S) = this proves that = or =1+ L,. On the other hand, we also have

I and

0. This implies that

= 0 and

as asserted.