Cryptography Reference
In-Depth Information
=
The joint probability distribution of two random variables
X
and
Y
.
Figure 4.4
4.2.2
Marginal Distributions
If
X
and
Y
are two random variables with joint probability distribution
P
XY
,then
the two
marginal distributions
P
X
and
P
Y
are defined as follows:
P
X
(
x
)=
y
P
XY
(
x, y
)
∈Y
P
Y
(
y
)=
x∈X
P
XY
(
x, y
)
Again, the notion of a marginal distribution can be generalized to more
than two random variables. If
X
1
,...,X
n
are
n
random variables with ranges
X
1
,...,
X
n
and joint probability distribution
P
X
1
...X
n
, then for any
m<n
the
marginal distribution
P
X
1
...X
m
is defined as follows:
P
X
1
...X
m
(
x
1
,...,x
m
)=
P
X
1
...X
n
(
x
1
,...,x
n
)
(
x
m
+1
,...,x
n
)
∈X
m
+1
...X
n
