This document provides an overview of one-dimensional random variables including definitions, types (discrete vs continuous), and probability distributions. It defines a random variable as a function that assigns a numerical value to each outcome of a random experiment. Random variables can be either discrete, taking on countable values, or continuous, assuming any value in an interval. The probability distribution of a discrete random variable is defined by a probability mass function, while a continuous random variable has a probability density function. Examples are given of both types of random variables and their distributions.

1. Page 1 of 11
CHAPTER THREE
3. ONE DIMENSIONAL RANDOM VARIABLES
3.1 Random variable: definition and distribution function
3.2 Discrete random variables
3.3 Continuous random variables
3.4 Cumulative distribution function and its properties
3.1 Random Variable
Introduction
The theory of probability distributions is widely used in both natural and social sciences. For that
matter, the concept of random variables and distributions is the backbone of statistics.
Particularly, the area of inferential statistics is entirely dependent on this concept. In
estimation and hypotheses testing, we extensively use some commonly known distributions.
Definition: Let E be an experiment and S be a sample space associated with the experiment. A
function X that assigns to every element ∈ , a real number, ( ), is called a random
variable.The set of all possible values of the random variable X is called the range space.
Moreover, the domain of the random variable X is S.
 Random variables are real valued functions that transform each sample point of a sample space to
real numbers. Hence, random variables serve as links between outcomes of a random experiment and
real numbers. Their significance can be seen from the fact that it is theoretically easier to deal
with numbers than outcomes.
 A random variable, X, is a function that assigns a single, but variable, value to each element
of a sample space.
 A random variable, X, provides a means of assigning numerical values to experimental
outcomes.

 A random variable, X, is a numerical description of the outcomes of the experiment or a
numerical valued function defined on sample space, usually denoted by capital letters.

2. Page 2 of 11
Example 1: Consider the different possible orderings of boy (B) and girl (G) in four sequential
births. There are 2*2*2*2= 16 possibilities, so the sample space is:
BBBB GGBB BGBB GBBB
BBBG BGBG GBBG GGBG
BBGB BGGB GBGB GGGB
BBGG BGGG GBGG GGGG
If girl and boy are each equally likely [P(G) = P(B) = 1/2], and the gender of each child is
independent of that of the previous child, then the probability of each of these 16 possibilities is:
(1/2)(1/2)(1/2)(1/2) = 1/16.
Now count the number of girls in each set of four sequential births:
BBBB (0) GGBB (2) BGBB (1) GBBB (1)
BBBG (1) BGBG (2) GBBG (2) GGBG (3)
BBGB (1) BGGB (2) GBGB (2) GGGB (3)
BBGG (2) BGGG (3) GBGG (3) GGGG (4)
Notice that:
 each possible outcome is assigned a single numeric value,
 all outcomes are assigned numeric values, and
 the value assigned varies across the outcomes.
The count of the number of girls is a random variable:

3. Page 3 of 11
Where RX is the range space of X and S is the sample space.
Example 2: Consider the experiment of tossing of a fair coin once.
The sample space is S= {H, T} where H denotes the outcome ‘Head’ and T denotes the outcome
‘Tail’. So, there are two possible outcomes H or T.
Now, let the random variable X represents the outcome `Head’, then X can take the value 0 or 1.
Example 3: Suppose a single fair die is rolled once.
The sample space of this experiment constitutes six possible outcomes, S = {1, 2, 3, 4, 5, 6}
Let the random variable X denotes the event: "anumber greater than 2 occurs’. Then the
random variable can assume the values 3, 4, 5 or 6.
Probability Distribution
The probability distribution for a random variable describes how the probabilities are distributed
over the values of the random variable.

4. Page 4 of 11
Depending upon the numerical values it can assume, a random variable can be classified into two
major divisions.
 Discrete Random Variable and
 Continuous Random Variable
3.2. Discrete Random Variable
Definition: Let X be a random variable. If the number of possible values of X (i.e. Rx) is finite
or countably infinite, then we call X as a discrete random variable. That is, the possible values of
X may be listed as , , … , ,… In the finite case, the list terminates and in the countably
infinite case the list continues indefinitely.
Examples:
• Toss a coin times and count the number of heads.
• Number of children in a family.
• Number of car accidents per week.
• Number of defective items in a given company.
• Number of bacteria per two cubic centimeter of water.
Probability Distribution of Discrete Random Variable
Definition:
: Let X be a discrete random variable. Hence, RX consists of at most a countably
infinite number of values, , , … with each possible outcome we associate a number
( ) = ( = ) called the probability of .The probabilities ( ) for = 1,2,3, . . . , , . ..
must satisfy the following two conditions:
I. ( ) ≥ 0 for all ,
II. ∑ ( ) = 1
The function p defined above is called the probability function (or point probability
function) of the random variable X. Furthermore, the tabular arrangement of all possible
values of X with their corresponding probability is called probability distribution of X.
Example 1: Construct a probability distribution for the number of heads in tossing of a fair coin
two times.
Types of Random Variables

5. Page 5 of 11
Solution: The sample space of the experiment contains the following:
S = {HH, HT, TH, TT}
Let the random variable X denotes the ‘number of heads’. We then use the probability function
P(X) to assign probability to each outcome consequently; the probability distribution is given
below:
0 1 2
( ) 1/4 1/2 1/4
Notes:
a) Suppose that the discrete random variable X may assume only a finite number of values,
say , , … , . If each outcome is equally probable, then we obviously have ( ) =
( ) = ⋯ = ( ) = .
b) If X assumes a countably infinite number of values, then it is impossible to have all
outcomes equally probable. For we cannot possibly satisfy the condition ∑ P(x ) = 1
if we must have ( ) = c for all .
c) In every finite interval, there will be at most afinite number of possible values of X. if
some such interval contains none of these possible values, we assign a probability zero to
it. That is, if = { , , … , } and if no ∈ [a, b], then ( ≤ ≤ ) = 0.
3.3. Continuous Random Variable
Definition: A random variable X is said to be a continuous random variable if it assumes all
values in some interval (c,d), where c and d are real numbers.
Examples:
• Height of students at a certain college.
• Mark of a student.
• Life time of light bulbs.
• Length of time required to complete a given training.
Probability density function ( ) of continuous random variable

6. Page 6 of 11
A random variable X is said to be a continuous random variable if there exists a function ,
called the probability density function (pdf) of X, satisfying the following conditions:
a) ( ) ≥ 0 for all
b) ∫ ( ) = 1
c) For any ℎ − ∞ < < < ∞, ℎ ( ≤ ≤ ) = ∫ ( )
Remark:
a. P(a≤x≤b) represents the area under the graph ( or curve) of the P.d.f of f between a and b.
b
b.
. It is a consequence of the above description of X that for any specified value of X, say X0,
w
we
e h
ha
av
ve
e p
p(
(X
X=
=X
X0
0)
)=
=0
0,
, s
si
in
nc
ce
e P
P(
(X
X=
=X
X0
0)
)=
= 0
)
(
0
0

 dx
x
f
x
x
W
We
e m
mu
us
st
t r
re
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al
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iz
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e,
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ho
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r,
, t
th
ha
at
t i
if
f w
we
e a
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ll
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w X
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to
o a
as
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va
al
lu
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es
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in
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in
nt
te
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th
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en
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z
ze
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ro
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ba
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il
li
it
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y d
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es
s n
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t i
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mp
pl
ly
y t
th
ha
at
t t
th
he
e e
ev
ve
en
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t i
is
s i
im
mp
po
os
ss
si
ib
bl
le
e.
. T
Th
ha
at
t i
is
s,
, z
ze
er
ro
o p
pr
ro
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ba
ab
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li
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ty
y
d
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oe
es
s n
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ce
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il
ly
y i
im
mp
pl
ly
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im
mp
po
os
ss
si
ib
bi
il
li
it
ty
y.
. H
He
en
nc
ce
e,
, i
in
n t
th
he
e c
co
on
nt
ti
in
nu
uo
ou
us
s c
ca
as
se
e P
P(
(A
A)
)=
=0
0 d
do
oe
es
s n
no
ot
t i
im
mp
pl
ly
y
A
A=
=
.
. I
In
n v
vi
ie
ew
w o
of
f t
th
hi
is
s f
fa
ac
ct
t,
, t
th
he
e f
fo
ol
ll
lo
ow
wi
in
ng
g p
pr
ro
ob
ba
ab
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il
li
it
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ie
es
s a
ar
re
e a
al
ll
l t
th
he
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sa
am
me
e i
if
f X
X i
is
s a
a c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m
v
va
ar
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ia
ab
bl
le
e.
.
P
P(
(c
c≤
≤X
X≤
≤d
d)
)=
= P
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(c
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)=
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(c
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)=
= P
P(
(c
c<
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d)
)
C
Ca
au
ut
ti
io
on
n:
: T
Th
he
e a
ab
bo
ov
ve
e p
pr
ro
op
pe
er
rt
ty
y c
ca
an
n n
ne
ev
ve
er
r b
be
e e
ex
xt
te
en
nd
de
ed
d t
to
o t
th
he
e c
ca
as
se
e o
of
f d
di
is
sc
cr
re
et
te
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
es
s.
.
E
Ex
xa
am
mp
pl
le
e 1
1:
: L
Le
et
t X
X b
be
e a
a c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e w
wi
it
th
h p
pr
ro
ob
ba
ab
bi
il
li
it
ty
y d
de
en
ns
si
it
ty
y f
fu
un
nc
ct
ti
io
on
n (
(P
P.
.d
d.
.f
f)
)


 


otherwise
0
1
0
if
2
)
(
x
x
x
f
a
a)
) V
Ve
er
ri
if
fy
y t
th
ha
at
t f
f i
is
s a
a P
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d.
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f b
b)
) f
fi
in
nd
d P
P(
(0
0.
.5
5<
<x
x<
<0
0.
.7
75
5)
)
Probability Density Function (pdf) of Continuous Random Variables

7. Page 7 of 11
a
a)
) A
A f
fu
un
nc
ct
ti
io
on
n f
f i
is
s a
a P
P.
.d
d.
.f
f i
if
ff
f t
th
he
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ol
ll
lo
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wi
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g t
tw
wo
o c
co
on
nd
di
it
ti
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ns
s a
ar
re
e s
sa
at
ti
is
sf
fi
ie
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d
i
i)
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f(
(x
x)
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0
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an
nd
d i
ii
i)
) 1
)
( 




dx
x
f
I
In
n o
ou
ur
r c
ca
as
se
e,
, i
it
t i
is
s e
ea
as
si
il
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y o
ob
bs
se
er
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va
ab
bl
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th
ha
at
t f
f (
(x
x)
) 
 0
0 b
be
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ca
au
us
se
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x i
is
s a
a p
po
os
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it
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ve
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um
mb
be
er
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or
r a
al
ll
l x
x
b
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et
tw
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ee
en
n 0
0 a
an
nd
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1.
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Mo
or
re
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ve
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r,
, f
fo
or
r o
ot
th
he
er
r v
va
al
lu
ue
es
s o
of
f x
x t
th
he
e g
gi
iv
ve
en
n f
fu
un
nc
ct
ti
io
on
n a
as
ss
su
um
me
es
s t
th
he
e v
va
al
lu
ue
e 0
0.
.
H
He
en
nc
ce
e t
th
he
e f
fi
ir
rs
st
t c
co
on
nd
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it
ti
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on
n i
is
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at
ti
is
sf
fi
ie
ed
d.
. W
We
e c
ca
an
n a
al
ls
so
o s
sh
ho
ow
w t
th
he
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se
ec
co
on
nd
d c
co
on
nd
di
it
ti
io
on
n a
as
s u
un
nd
de
er
r.
.
1
0
1
0
0dx
2
0
f(x)dx
)
(
)
(
)
(
1
0 1
0
1
0 1
0









 
 

 










xdx
dx
dx
x
f
dx
x
f
dx
x
f
T
Th
he
er
re
ef
fo
or
re
e,
, w
we
e c
ca
an
n c
co
on
nc
cl
lu
ud
de
e t
th
ha
at
t t
th
he
e g
gi
iv
ve
en
n f
fu
un
nc
ct
ti
io
on
n i
is
s a
a P
P.
.d
d.
.f
f.
.
b
b)
) P
P (
(0
0.
.5
5<
<x
x<
<0
0.
.7
75
5)
)=
=
16
5
4
1
16
9
x
2
75
.
0
5
.
0
2
75
.
0
5
.
0




 xdx
E
Ex
xa
am
mp
pl
le
e 2
2:
: S
Su
up
pp
po
os
se
e t
th
ha
at
t X
X i
is
s a
a c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e w
wi
it
th
h P
P.
.d
d.
.f
f


 


otherwise
0
1
x
0
if
3x
)
(
2
x
f
a
a)
) V
Ve
er
ri
if
fy
y t
th
ha
at
t f
f i
is
s a
a P
P.
.d
d.
.f
f b
b)
) P
P(
(X
X>
>1
1/
/3
3)
) c
c)
) P
P(
(1
1/
/4
4 
 x
x 
 2
2/
/3
3)
)
C
Co
on
nd
di
it
ti
io
on
na
al
l P
Pr
ro
ob
ba
ab
bi
il
li
it
ty
y i
in
n t
th
he
e C
Ca
as
se
e o
of
f C
Co
on
nt
ti
in
nu
uo
ou
us
s R
Ra
an
nd
do
om
m V
Va
ar
ri
ia
ab
bl
le
es
s
R
Re
ec
ca
al
ll
l t
th
ha
at
t f
fo
or
r t
th
he
e c
ca
as
se
e o
of
f d
di
is
sc
cr
re
et
te
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
es
s t
th
he
e c
co
on
nd
di
it
ti
io
on
na
al
l p
pr
ro
ob
ba
ab
bi
il
li
it
ty
y i
is
s g
gi
iv
ve
en
n b
by
y
)
(
)
(
)
/
(
B
P
B
A
P
B
A
P


T
Th
he
e s
sa
am
me
e f
fo
or
rm
mu
ul
la
at
ti
io
on
n h
ho
ol
ld
ds
s t
tr
ru
ue
e f
fo
or
r t
th
he
e c
ca
as
se
e o
of
f c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
es
s e
ex
xc
ce
ep
pt
t t
th
ha
at
t A
A a
an
nd
d B
B
t
ta
ak
ke
e r
ra
an
ng
ge
e o
of
f v
va
al
lu
ue
es
s (
(i
in
nt
te
er
rv
va
al
ls
s)
) r
ra
at
th
he
er
r t
th
ha
an
n d
di
is
sc
cr
re
et
te
e v
va
al
lu
ue
es
s.
.
E
Ex
xa
am
mp
pl
le
e:
: T
Th
he
e d
di
ia
am
me
et
te
er
r o
of
f a
an
n e
el
le
ec
ct
tr
ri
ic
c c
ca
ab
bl
le
e i
is
s a
as
ss
su
um
me
ed
d t
to
o b
be
e a
a c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e,
, s
sa
ay
y
X
X,
, w
wi
it
th
h P
P.
.d
d.
.f
f


 



otherwise
0
1
0
if
)
1
(
6
)
(
x
x
x
x
f
C
Co
om
mp
pu
ut
te
e 










3
2
3
1
2
1
x
x
P
Solution

8. Page 8 of 11
3.4 Cumulative distribution function and its properties
D
De
ef
fi
in
ni
it
ti
io
on
n (
(C
Cu
um
mu
ul
la
at
ti
iv
ve
e D
Di
is
st
tr
ri
ib
bu
ut
ti
io
on
n F
Fu
un
nc
ct
ti
io
on
n)
) L
Le
et
t X
X b
be
e a
a r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e (
(d
di
is
sc
cr
re
et
te
e o
or
r
c
co
on
nt
ti
in
nu
uo
ou
us
s)
).
. T
Th
he
e f
fu
un
nc
ct
ti
io
on
n F
F t
th
ha
at
t i
is
s d
de
ef
fi
in
ne
ed
d a
as
s
F
F(
(x
x)
) =
= P
P[
[X
X
x
x]
]
i
is
s c
ca
al
ll
le
ed
d t
th
he
e c
cu
um
mu
ul
la
at
ti
iv
ve
e d
di
is
st
tr
ri
ib
bu
ut
ti
io
on
n f
fu
un
nc
ct
ti
io
on
n (
(C
Cd
df
f)
) o
of
f t
th
he
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e X
X.
.
N
No
ot
ta
at
ti
io
on
n:
: C
Cu
um
mu
ul
la
at
ti
iv
ve
e d
di
is
st
tr
ri
ib
bu
ut
ti
io
on
n f
fu
un
nc
ct
ti
io
on
n i
is
s u
us
su
ua
al
ll
ly
y d
de
en
no
ot
te
ed
d b
by
y u
up
pp
pe
er
r c
ca
as
se
e a
al
lp
ph
ha
ab
be
et
ts
s.
.
T
Th
he
eo
or
re
em
ms
s
1
1.
. I
If
f X
X i
is
s a
a d
di
is
sc
cr
re
et
te
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e,
, t
th
he
en
n
x
x
p
x
F
j
j 
  j
x
all
for
)
(
)
(
2
2.
. I
If
f X
X i
is
s a
a c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e,
, t
th
he
en
n

x


 ds
s
f
x
F )
(
)
(
E
Ex
xa
am
mp
pl
le
e 1
1:
: L
Le
et
t S
S=
={
{H
H,
,T
T}
} a
an
nd
d X
X:
: t
th
he
e n
nu
um
mb
be
er
r o
of
f h
he
ea
ad
ds
s.
.
C
Cl
le
ea
ar
rl
ly
y,
, X
X i
is
s a
a d
di
is
sc
cr
re
et
te
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e.
. S
Si
in
nc
ce
e P
P(
(x
x=
=0
0)
)=
=1
1/
/2
2=
=P
P(
(x
x=
=1
1)
),
, w
we
e h
ha
av
ve
e










1
x
if
1
1
x
0
if
2
/
1
0
X
if
0
)
(x
F
E
Ex
xa
am
mp
pl
le
e 2
2:
: S
Su
up
pp
po
os
se
e t
th
ha
at
t a
a r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e X
X a
as
ss
su
um
me
es
s t
th
he
e s
si
ix
x v
va
al
lu
ue
es
s 1
1,
, 2
2,
, 3
3,
, 4
4,
, 5
5,
, 6
6 w
wi
it
th
h
p
pr
ro
ob
ba
ab
bi
il
li
it
ty
y 1
1/
/6
6.
. T
Th
he
en
n,
,
























6
x
if
1
6
x
5
if
5/6
5
x
4
if
4/6
4
x
3
if
3/6
3
2
if
2/6
2
x
1
if
6
/
1
1
X
if
0
)
(
x
x
F

9. Page 9 of 11
E
Ex
xa
am
mp
pl
le
e 3
3:
: F
Fi
in
nd
d t
th
he
e c
cd
df
f o
of
f t
th
he
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e X
X w
wh
ho
os
se
e P
P.
.d
d.
.f
f i
is
s g
gi
iv
ve
en
n b
by
y











Otherwise
0
2
x
1
if
2
1
x
0
if
x
)
( x
x
f
S
So
ol
lu
ut
ti
io
on
n:
:

















2
2
1
1
0
0
)
(
)
(
)
(
)
(
)
(
)
(
F(x) dt
t
f
dt
t
f
dt
t
f
dt
t
f
dt
t
f
x
X
P
x
C
Ca
as
se
e 1
1:
: F
Fo
or
r x
x
0
0
0
0
)
(
F(x) 
 






x
x
dt
dt
t
f
C
Ca
as
se
e 2
2:
: F
Fo
or
r 0
0
 x
x 
 1
1
2
2
t
0
)
(
)
(
)
(
)
(
F(x)
2
0
2
0
0
0
x
tdt
dt
t
f
dt
t
f
dt
t
f
x
X
P
x
x
x
x








 


 



C
Ca
as
se
e 3
3:
: F
Fo
or
r 1
1
 x
x 
 2
2
1
2
2
2
2
2
1
2
2
t
0
)
(
)
(
)
(
)
(
)
(
F(x)
2
1
2
1
1
0
2
1
0 1
0























 
 

 



x
x
x
x
x
dt
t
f
dt
t
f
dt
t
f
dt
t
f
x
X
P
x
x
x
x
C
Ca
as
se
e 4
4:
: F
Fo
or
r x
x>
>2
2
1
0
2
3
2
2
1
0
2
2
2
t
0
)
(
)
(
)
(
)
(
)
(
)
(
F(x)
2
2
1
2
1
0
2
2
1
0
2
1
0

































 

 



x
x
x
dt
x
x
dt
t
f
dt
t
f
dt
t
f
dt
t
f
dt
t
f
x
X
P
T
Th
he
er
re
ef
fo
or
re
e,
, t
th
he
e c
cd
df
f o
of
f X
X b
be
ec
co
om
me
es
s,
,

10. Page 10 of 11
















2
x
if
1
2
x
1
if
1
-
2
x
-
2x
1
x
0
if
2
0
x
if
0
)
( 2
2
x
x
F
E
Ex
xe
er
rc
ci
is
se
es
s:
:
1
1.
. F
Fi
in
nd
d t
th
he
e c
cd
df
f o
of
f a
a r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e X
X w
wh
ho
os
se
e P
P.
.d
d.
.f
f i
is
s g
gi
iv
ve
en
n b
by
y


 

Otherwise
0
0
x
if
e
)
(
-x
x
f
2
2.
. D
De
et
te
er
rm
mi
in
ne
e t
th
he
e c
cd
df
f o
of
f t
th
he
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e X
X w
wi
it
th
h P
P.
.d
d.
.f
f



















Otherwise
0
3
2
if
2
3
2
1
-
2
x
1
if
2
1
1
x
0
if
x
2
1
)
(
x
x
x
f
P
Pr
ro
op
pe
er
rt
ti
ie
es
s o
of
f C
Cu
um
mu
ul
la
at
ti
iv
ve
e D
Di
is
st
tr
ri
ib
bu
ut
ti
io
on
n F
Fu
un
nc
ct
ti
io
on
n (
(c
cd
df
f)
) F
F
a
a)
) T
Th
he
e f
fu
un
nc
ct
ti
io
on
n F
F i
is
s a
a n
no
on
n-
-d
de
ec
cr
re
ea
as
si
in
ng
g f
fu
un
nc
ct
ti
io
on
n,
,
b
b)
) 1
)
(
lim
and
0
)
(
lim
x
x






x
F
x
F
T
Th
he
eo
or
re
em
m
I
If
f X
X i
is
s a
a c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e,
, t
th
he
en
n w
we
e h
ha
av
ve
e )
(
)
( x
F
dx
d
x
f  .
. O
On
n t
th
he
e o
ot
th
he
er
r h
ha
an
nd
d,
, i
if
f X
X i
is
s a
a
d
di
is
sc
cr
re
et
te
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e,
, t
th
he
en
n w
we
e h
ha
av
ve
e )
(
)
(
)
( 1


 j
j
j x
F
x
F
x
P .
.
P
Pr
ro
oo
of
f
C
Ca
as
se
e 1
1:
: X
X i
is
s a
a c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e
calculus
of
theorem
l
fundamenta
the
.......by
)
(
)
(
dx
d
)
(
dx
d
)
(
)
(
x
f
dt
t
f
x
F
dt
t
f
x
F
x
x










C
Ca
as
se
e 2
2:
: X
X i
is
s a
a d
di
is
sc
cr
re
et
te
e r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e

11. Page 11 of 11
)
(
)
(
)
(
)
(
)
(
)
(
...
)
(
)
(
)
(
)
(
)
(
)
(
...
)
(
)
(
)
(
)
(
1
1
2
2
1
1
1
1
2
1
x
f
x
P
x
F
x
F
x
P
x
P
x
P
x
P
x
x
P
x
F
x
P
x
P
x
P
x
P
x
x
P
x
F
j
j
j
j
j
j
j
j
j
j
j
























R
Re
em
ma
ar
rk
k:
: P
P(
(a
a
 X
X 
 b
b)
)=
=F
F(
(b
b)
)-
-F
F(
(a
a)
)
E
Ex
xa
am
mp
pl
le
e:
: S
Su
up
pp
po
os
se
e X
X i
is
s a
a c
co
on
nt
ti
in
nu
uo
ou
us
s r
ra
an
nd
do
om
m v
va
ar
ri
ia
ab
bl
le
e w
wi
it
th
h c
cd
df
f






0
x
if
e
-
1
0
x
if
0
)
( x
-
x
F
F
Fi
in
nd
d t
th
he
e P
P.
.d
d.
.f
f o
of
f X
X.
.