Information Technology Reference
In-Depth Information
As for the probability density function, we have similar results to those for prob-
abilities. For example, we have
p(x,y)
p(x)
p(y
|
x)
=
.
21.1.2.3 Popular Distributions
There are many well-studied probability distributions. In this subsection, we list a
few of them for example.
•
p
x
(
1
p)
1
−
x
,x
Bernoulii distribution:
P(X
=
x)
=
−
=
0or1.
λ)λ
k
exp
(
−
•
Poisson distribution:
P(X
=
k)
=
.
k
!
μ)
2
2
σ
2
)
.
(x
−
1
•
Gaussion distribution:
p(x)
=
√
2
πσ
exp
(
−
•
Exponential distribution:
p(x)
=
λ
exp
(
−
λx), x
≥
0.
21.1.3 Expectations and Variances
Expectations and variances are widely used concepts in probability theory. Expec-
tation is also referred to as mean, expected value, or first moment. The expectation
of a random variable, denoted as
E
[
X
]
, is defined by
E
[
X
]=
aP(X
=
a),
for the discrete case;
a
∞
E
[
X
]=
xp(x) dx,
for the continuous case.
−∞
When the random variable
X
is an indicator variable (i.e., it takes a value from
{
0
,
1
}
), we have the following result:
E
[
X
]=
P(X
=
1
).
The expectation of a random variable has the following three properties.
•
[
X
1
+
X
2
+···+
X
n
]=
E
[
X
1
]+
E
[
X
2
]+···+
E
[
X
n
]
.
E
•
E
[
XY
]=
E
[
X
]
E
[
Y
]
,if
X
and
Y
are independent.
•
.
The variance of a distribution is a measure of the spread of it. It is also referred
to as the second moment. The variance is defined by
If
f
is a convex function,
f(E
[
X
]
)
≤
E
[
f(x)
]
E
X
]
2
.
Va r
(X)
=
−
E
[
X
The variance of a random variable is often denoted as
σ
2
.And
σ
is called the
standard deviation.
The variance of a random variable has the following properties.
