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.