Geoscience Reference
In-Depth Information
2.3.3 Sum of Two Random Variables
Other frequency distribution models include the uniform or rectangular distribution.
It can be defined as:
1
fx
ðÞ
¼
a
if
a
<
x
<
b
;
fx
ðÞ
¼
0 otherwise
b
This distribution is shown graphically in Fig.
2.2
for
a
¼0 and
b
¼5. Suppose
that the normal distribution is written in standard form
(
x
). It is shown in Fig.
2.2
as well. A more general problem of mathematical statistics that has been discussed
by Wilks (
1962
) and Gnedenko (
1963
) and at a more advanced level by Lo`ve
(
1963
) is: What is the frequency distribution of the sum of two random variables?
ˆ
Box 2.6: Sum of Uniform and Normal Random Variables
Suppose that
X
1
and
X
2
are two random variables and that their sum is written
as
Y
X
1
+
X
2
with frequency distribution function
f
(
y
). The joint probability
P
(
X
1
¼
¼
x
1
and
X
2
¼
x
2
) can be represented as
f
(
x
1
,
x
2
) for infinitesimal area
ZZ
dx
1
dx
2
, and:
PY
ð
<
y
Þ
¼
Fy
ðÞ
¼
fx
1
;
ð
x
2
Þ
dx
1
dx
2
. Integrating the
x
1
þ
x
2
<
y
joint probability of
X
1
and
X
2
over the area where
x
1
+
x
2
<
y
provides the total
probability
that
Y
is
less
than
y
¼
x
1
+
x
2
.
It
follows
that:
dx
1
. After some manipulation, it can be
Z
yx
1
Z
yx
1
Fy
ðÞ
¼
fx
1
;
ð
x
2
Þ
dx
2
1
1
¼
R
1
1
derived that
f
(
y
)
x
)
f
1
(
x
)
dx
.If
h
(
x
) represents the sum of the
uniform distribution with
f
(
x
) as defined at the beginning of this section
and
f
2
(
y
the
standard
normal
ˆ
(
x
),
it
follows
that
Z
b
a
ˆ
1
1
hx
ðÞ
¼
ð
x
y
Þ
dy
¼
a
ʦ
½
ð
x
a
Þʦ
ð
x
b
Þ
.
b
a
b
5)]. Single values of
h
(
x
)
were calculated and connected by a smooth curve in this figure. For example,
if
x
For the example of Fig.
2.5
,
h
(
x
)
¼
0.2 · [
ʦ
(
x
)
ʦ
(
x
0.1952. The three
curves shown in Fig.
2.2
each have total area under the curve equal to one.
The curve for
h
(
x
) resembles a Gaussian curve so that it would be difficult to
distinguish it from a Gaussian curve on the basis of a sample of
n
values unless
n
is
large. In the example, the base of the uniform distribution
f
(
x
) is five times the
standard deviation of the normal curve
¼
2,
h
(
x
)
¼
0.2 · [
ʦ
(2)
ʦ
(3)]
¼
0.2 · [0.9773
0.0014]
¼
(
x
). In fact, this standard deviation was
used as unit of distance along the
X
-axis. The shape of
h
(
x
) will change if the ratio
(base of
f
/standard deviation of
ˆ
) is changed. If this ratio is decreased,
h
(
x
)
approaches the normal form. On the other hand, if it is increased,
h
(
x
) begins to
develop a flat top. Its shape then approaches that of
f
(
x
). More mathematical details
ˆ
Search WWH ::
Custom Search