Geoscience Reference
In-Depth Information
A.2 Random Variables
Functions employed to translate probabilities p in (S,
ʛ
) to (R, B) are called random
variables. A random variable in (S,
) is a function X with domain S such that the
inverse image by X of any interval is an event in
ʛ
.
For any random variable X, the probability p
X
in (R, B) determined by
ʛ
X
1
p
ð
ð
a
;
b
Þ
¼
p
ð
ð
ð
a
;
b
ÞÞ
X
is called the probability distribution of X and the cumulative probability function,
F
X
de
ned by
X
1
F
X
x
ðÞ
¼
p
ð
ðð/;
x
ÞÞ
is called the cumulative distribution of X. If F
X
has a density, this is called also the
density of X.
Two main types of random variables are relevant: those with probabilities p
X
for
which there is a countable set S
p
={x
i
}
i
∈
N
of values for which p
X
(S
p
) = 1 and those
with an absolutely continuous cdf. In the
first type, named discrete, the probability
distribution of X is determined by {p
X
(x
i
)}
i
∈
N
. In the second, it is determined by its
density.
To denote events in the Borel sigma algebra determined by values of a random
variable X de
, p) is employed a proper notation:
for a phrase r(X) involving X, [r(X)] denotes the set {s
ned on a probability space (S,
ʛ
∈
S| r(X(s))} and p[f(X)]
denotes the probability of the set of elements s of the sample space S for which the
phrase f(X) is true when X is replaced by the real number X(s).
A.3 Expected Values
The expected value of a random variable X of
the discrete type with
P
i
2
N
p
X
f
ð
x
i
g
Þ
¼
1is
X
E
ð
X
Þ
¼
x
i
p
X
x
ðÞ:
i
2
N
For a random variable X with density f
X
, the expected value of X is
Z
xf
X
x
ðÞ
¼
ðÞ
:
EX
dx
This sum and this integral may naturally not converge. So, for some variables,
the expected value is not de
ned.
As a weighted average of the possible numerical values assumed by the random
variable, with weights given by the probability, the expected value, like a centre of
mass in a bar, is a number around which the probability distribution is spread. For
this reason it is called a location parameter.
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