Geoscience Reference
In-Depth Information
0.2
Fig. 13.1 Example of a rod graph,
showing the relative
frequency of a discrete
random variable
X
for different values
x
i
=
...,−
2
, −
1
,
0
,
1
,
etc.
0.1
0
−
5
0
5
10
15
x
i
in which the symbol
P
denotes the probability of the event stated between the curly
brackets. A rod graph is a natural way to represent the distribution of the probabilities
as relative frequencies of a discrete variable (see Figure 13.1).
A random variable is called continuous when it can assume any value
x
in a certain
range of real numbers, which may or may not be unbounded. In this case, since the
variable
X
can assume an infinity of values, it is more appropriate to consider the
probability that it is smaller than or equal to a given value
x
. This defines the (
probability
)
distribution function
as
F
(
x
)
=
P
{
X
≤
x
}
(13.3)
where again
P
is the probability of the event between the curly brackets. For all values
of
x
, where the distribution function is smooth, the (
probability
)
density function
can be
defined by
{}
dF
(
x
)
dx
f
(
x
)
=
(13.4)
Thus the probability that
X
≤
x
can also be written in terms of the density function as
x
P
{
X
≤
x
}=
f
(
y
)
dy
(13.5)
−∞
in which
y
is the dummy variable of integration. This is illustrated in Figure 13.2. In the
same way the probability, that the random variable occur in a certain range
x
1
<
X
≤
x
2
,
is given by
x
2
F
(
x
2
)
−
F
(
x
1
)
=
f
(
x
)
dx
(13.6)
x
1
