Cryptography Reference
In-Depth Information
=
Figure 4.3
The probability distribution of a random variable
X
.
The joint probability distribution of two random variables
X
and
Y
is illus-
trated in Figure 4.4. Some events from the sample space Ω (on the left side) are
mapped to
x
(on the right side), and the probability that
x
and
y
occur as maps is
P
(
X
=
x, Y
=
y
)=
P
XY
(
x, y
).
Similarly, for
n
random variables
X
1
,...,X
n
(with ranges
∈X
and
y
∈Y
X
n
), one
can compute the probability that
X
i
takes on the value
x
i
∈X
i
for
i
=1
,...,n
.In
fact, the
joint probability distribution
of
X
1
,...,X
n
(i.e.,
P
X
1
...X
n
) is a mapping
from
X
1
,...,
+
that is formally defined as follows:
X
1
×
...
×X
n
to
R
+
P
X
1
...X
n
:
X
1
×
...
×X
n
−→
R
(
x
1
,...,x
n
)
−→
P
X
1
...X
n
(
x
1
,...,x
n
)=
P
(
X
1
=
x
1
,...,X
n
=
x
n
)=
Pr[
ω
]
ω∈
Ω:
X
1
(
ω
)=
x
1
;
...
;
X
n
(
ω
)=
x
n
The joint probability distribution of
n
random variables
X
1
,...,X
n
is il-
lustrated in Figure 4.5. Some events from the sample space Ω (on the left side)
are mapped to
x
1
∈X
1
, ..., x
n
∈X
n
(on the right side), and the probabil-
ity that
x
1
,...,x
n
actually occur as maps is
P
(
X
1
=
x
1
,...,X
n
=
x
n
)=
P
X
1
...X
n
(
x
1
,...,x
n
).
