B. Distribution Function
1. Definition of cumulative distribution function
Definition 4.2
The cumulative distribution function cdf, or simply distribution function, F of a random variable X is defined as
F b = P{X
b}
b .
Notesμ here “
” means “for all”ν b is a real numberν and the notation “{X b}” is considered as an event as mentioned before.
Example 4.3
In a game of tossing a fair coin, let X be a random variable with its value defined as +5 winning 5 dollars if a head H appears and as
3 losing 3 dollars if a tail T appears. Derive the cdf Fb of X and draw a diagram for it.
Solution:
The random variable X takes only two discrete values of 3 and +5.
Concept used to derive Fb: enumerate all cases for b. For b
F b = P{X
b} = P neither outcome T nor H will “yield” any value of X
b +5, Fb = P{X b} = P{T} = 12 because only the value of X =
b +5. Note: here, the notation T in P{T} above is considered as an event including just
an element, namely, the set {T}. Similar interpretations will be applied to subsequent discussions.
For +5 b, Fb = P{X b} = P{T, H} = 1 because X = 3 and +5 when T and
b.
A cdf diagram for the random variable X is shown in Fig. 4.1. Note the continuity condition at the discrete point of b =
3 or +5.
2. Notes about limit points ---
In Fig. 4.1, the hollow circle at the right end of the middle line segment for Fb = 12, for example, means the “limit point” 5 , which is the largest real value smaller
than 5 .
Formally, the limit point b
is defined as
lim
n
b
1 n
and may be regarded to be located right
to the left of the point at b.
Similarly, b
+
is defined as
lim
n
b +
1 n
. Such points do not appear in Fig. 4.1.
Fig. 4.1 Cumulative distribution function Fb for Example 4.3.
3. Some properties of the cdf
The following properties are intuitive in general can be seen to be true from Fig. 4.1 above; for proof, see the reference book, or prove them by yourself.
Property 4.1 ---
F a
Fb if a b, i.e., F is a nondecreasing function.
Property 4.2 ---
lim
b
F b = 1.
Property 4.3 ---
lim
b
F b = 0.
Property 4.4 ---
For any b
and any decreasing sequence b
1
, b
2
, b
3
, … which converges to b, it is true that
lim
n
F b
n
= Fb i.e., F is right continuous.
A note: Fb with b = 5, for example for Fig. 4.1 above, is just denoted by the solid circle at the left end of the right line segment for Fb = 1.
4. Some facts
All probability questions can be answered in terms of the cdf. Some examples are the following facts.
Fact 4.1 ---
P {a X
b} = Fb F
a.
3
b
F b
Proof: easy to prove from the definition of the cdf, and the fact {X
b} = {a X b
}U{X a} where {a X b} and {X a} are mutually exclusive.
Fact 4.2 ---
P {X b} = Fb
noteμ there is no sign of “=” in X b. Proof:
left as an exercise.
Fact 4.3 ---
P {X = b} = Fb
Fb
noteμ this value is the “jump from b
to b ”.
Proof: left as an exercise.
Fact 4.4 ---
P {X b} = 1
Fb. Proof:
left as an exercise.
Note: Fb
= P{X b} P{X b} = Fb. An example can be seen from Fig. 4.1
where 12 = F5
F5 = 1.
Example 4.4
Given a cdf as follows as illustrated by Fig. 4.2: F
x = 0 for x 0;
A = x2
for 0 x 1;
B = 23
for 1 x 2;
C = 1112
for 2 x 3;
D = 1
for 3 x,
E compute the values P{2 X
4}, P{X 3}, P{X = 1}, and P{X 12}. Solution:
P {2 X
4} = F F
2 by Fact 4.1
P {X 3} = F3 = 1112.
by Fact 4.2 and D above
P {X = 1} = P{X
1} P{X 1} by Fact 4.3
= F F
1
by definition and Fact 4.2 = 23 12
by C and B above = 16.
P {X
P {X
F
12 = 34. by Fact 4.4
Fig. 4.2 Cumulative distribution function Fx for Example 4.4
C. Discrete Random Variable
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