In the theoretical discussion on Random Variables and Probability, we note that
the probability distribution induced by
a random variable X is determined uniquely by a consistent assignment of mass to semiinfinite
intervals of the form (∞,t](∞,t] for each real t. This suggests that a natural description
is provided by the following.
Definition
The distribution function F_{X} for random variable X is given by
F
X
(
t
)
=
P
(
X
≤
t
)
=
P
(
X
∈
(

∞
,
t
]
)
∀
t
∈
R
F
X
(
t
)
=
P
(
X
≤
t
)
=
P
(
X
∈
(

∞
,
t
]
)
∀
t
∈
R
(1)In terms of the mass distribution on the line, this is the probability mass at or to
the left of the point t. As a consequence, F_{X} has the following properties:
 (F1) : F_{X} must be a nondecreasing function, for if t>st>s there must be at least
as much probability mass at or to the left of t as there is for s.
 (F2) : F_{X} is continuous from the right, with a jump in the amount p_{0} at
t_{0} iff
P(X=t0)=p0P(X=t0)=p0. If the point t approaches t_{0} from the left, the interval
does not include the probability mass at t_{0} until t reaches that value, at which point the
amount at or to the left of t increases ("jumps") by amount p_{0}; on the other hand, if t approaches t_{0}
from the right, the interval includes the mass p_{0} all the way to and including t_{0}, but drops
immediately as t moves to the left of t_{0}.
 (F3) : Except in very unusual cases involving random variables which may take “infinite”
values, the probability mass included in (∞,t](∞,t] must increase to one as
t moves to the right; as t moves to the left, the probability mass included must decrease
to zero, so that
FX(∞)=limt→∞FX(t)=0andFX(∞)=limt→∞FX(t)=1FX(∞)=limt→∞FX(t)=0andFX(∞)=limt→∞FX(t)=1
(2)
A distribution function determines the probability mass in each semiinfinite interval
(∞,t](∞,t]. According to the discussion referred to above, this determines uniquely
the induced distribution.
The distribution function F_{X} for a simple random variable is easily visualized. The
distribution consists of point mass p_{i} at each point t_{i} in the range. To the left of
the smallest value in the range, FX(t)=0FX(t)=0; as t increases to the smallest value t_{1},
FX(t)FX(t) remains constant at zero until it jumps by the amount p_{1}.. FX(t)FX(t) remains constant
at p_{1} until t increases to t_{2}, where it jumps by an amount p_{2} to the value p1+p2p1+p2. This continues until the value of FX(t)FX(t)reaches 1 at the largest value t_{n}. The
graph of F_{X} is thus a step function, continuous from the right, with a jump in the amount
p_{i} at the corresponding point t_{i} in the range. A similar situation exists for a discretevalued
random variable which may take on an infinity of values (e.g., the geometric distribution
or the Poisson distribution considered below). In this case, there is always some probability
at points to the right of any t_{i}, but this must become vanishingly small as t increases,
since the total probability mass is one.
The procedure ddbn may be used to plot the distributon function for a simple
random variable from a matrix X of values and a corresponding matrix PXPX of
probabilities.
>> c = [10 18 10 3]; % Distribution for X in Example 6.5.1
>> pm = minprob(0.1*[6 3 5]);
>> canonic
Enter row vector of coefficients c
Enter row vector of minterm probabilities pm
Use row matrices X and PX for calculations
Call for XDBN to view the distribution
>> ddbn % Circles show values at jumps
Enter row matrix of VALUES X
Enter row matrix of PROBABILITIES PX
% Printing details See Figure 1