In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one.
The positions (indices) of a probability vector represent the possible outcomes of a discrete random variable, and the vector gives us the probability mass function of that random variable, which is the standard way of characterizing a discrete probability distribution.
Here are some examples of probability vectors. The vectors can be either columns or rows.
x
0
=
[
0.5
0.25
0.25
]
,
x
1
=
[
0
1
0
]
,
x
2
=
[
0.65
0.35
]
,
x
3
=
[
0.3
0.5
0.07
0.1
0.03
]
.
Writing out the vector components of a vector
p
as
p
=
[
p
1
p
2
⋮
p
n
]
or
p
=
[
p
1
p
2
⋯
p
n
]
the vector components must sum to one:
∑
i
=
1
n
p
i
=
1
Each individual component must have a probability between zero and one:
0
≤
p
i
≤
1
for all
i
. Therefore the set of stochastic vectors coincides with the standard
(
n
−
1
)
-simplex. It is a point if
n
=
1
, a segment if
n
=
2
, a (filled) triangle if
n
=
3
, a (filled) tetrahedron
n
=
4
, etc.
The mean of any probability vector is
1
/
n
.
The shortest probability vector has the value
1
/
n
as each component of the vector, and has a length of
1
/
n
.
The longest probability vector has the value 1 in a single component and 0 in all others, and has a length of 1.
The shortest vector corresponds to maximum uncertainty, the longest to maximum certainty.
The length of a probability vector is equal to
n
σ
2
+
1
/
n
; where
σ
2
is the variance of the elements of the probability vector.
