Cryptography Reference
In-Depth Information
P
X|Y
=
y
(
x
)=
P
XY
(
x, y
)
P
Y
(
y
)
whenever
P
Y
(
y
)
>
0. More specifically, the conditional probability distribution
P
X|Y
of
X
given
Y
is a two-argument function that is defined as follows:
+
P
X|Y
:
X×Y −→
R
(
x, y
)
−→
P
X|Y
(
x, y
)=
P
(
X
=
x
|
Y
=
y
)=
P
(
X
=
x, Y
=
y
)
P
(
Y
=
y
)
=
P
XY
(
x, y
)
P
Y
(
y
)
Note that the two-argument function
P
X|Y
(
·
,
·
) is not a probability distribution
on
X×Y
, but that for every
y
∈Y
, the one-argument function
P
X|Y
(
·
,y
) is
a probability distribution, meaning that
x∈X
P
X|Y
(
x, y
) mustsumupto1for
every
y
with
P
Y
(
y
)
>
0. Also note that
P
X|Y
(
x, y
) is defined only for values with
P
(
Y
=
y
)=
P
Y
(
y
)
=0.
4.2.4
Expectation
The expectation of a random variable gives some information about its order of
magnitude, meaning that if the expectation is small (large), then large (small) values
are unlikely to occur. More formally, let
X
:Ω
→X
be a random variable and
X
be a finite subset of the real numbers (i.e.,
). Then the
expectation
or
mean
of
X
(denoted as
E
[
X
]) can be computed as follows:
X⊂
R
E
[
X
]=
x∈X
x
=
x∈X
Pr[
X
=
x
]
·
P
X
(
x
)
·
x
(4.2)
The expectation is best understood in terms of betting. Let us consider the
situation in which a person is playing a game in which one can win one dollar or lose
two dollars. Furthermore, there is a 2
/
3 probability of winning, a 1
/
6 probability
of losing, and a 1
/
6 probability of a draw. This situation can be modeled using a
discrete probability space with a sample space Ω=
(with
W
standing for
“win,”
L
standing for “lose,” and
D
standing for “draw”) and a probability measure
that assigns Pr[
W
]=2
/
3 and Pr[
L
]=Pr[
D
]=1
/
6. Against this background, the
random variable
X
can be used to specify wins and losses—
X
(
W
)=1,
X
(
L
)=2,
{
W, L, D
}
