Cryptography Reference
In-Depth Information
Pr[
X
= 10]
=
Pr[
X
=14
−
10] = Pr[
X
=4]=3
/
36
Pr[
X
= 11]
=
Pr[
X
=14
−
11] = Pr[
X
=3]=2
/
36
Pr[
X
= 12]
=
Pr[
X
=14
−
12] = Pr[
X
=2]=1
/
36
It can easily be verified that all probabilities sum up to one:
1
36
+
2
36
+
3
36
+
4
36
+
5
36
+
6
36
+
5
36
+
4
36
+
3
36
+
2
36
+
1
36
=
36
36
=1
We next look at some probability distributions of random variables.
4.2.1
Probability Distributions
If
X
:Ω
→X
is a random variable with sample space Ω and range
X
, then the
+
. It is formally
probability distribution
of
X
(i.e.,
P
X
) is a mapping from
X
to
R
defined as follows:
+
P
X
:
X−→
R
x
−→
P
X
(
x
)=
P
(
X
=
x
)=
Pr[
ω
]
ω
∈
Ω:
X
(
ω
)=
x
The probability distribution of a random variable
X
is illustrated in Figure 4.3.
Some events from the sample space Ω (on the left side) are mapped to
x
(on
the right side), and the probability that
x
occurs as a map is
P
(
X
=
x
)=
P
X
(
x
).
It is possible to define more than one random variable for a discrete random
experiment. If, for example,
X
and
Y
are two random variables with ranges
∈X
X
and
Y
,then
P
(
X
=
x, Y
=
y
) refers to the probability that
X
takes on the value
x
∈X
and
Y
takes on the value
y
. Consequently, the
joint probability distribution
of
X
and
Y
(i.e.,
P
XY
) is a mapping from
∈Y
+
. It is formally defined as
X×Y
to
R
follows:
+
P
XY
:
X×Y −→
R
(
x, y
)
−→
P
XY
(
x, y
)=
P
(
X
=
x, Y
=
y
)=
Pr[
ω
]
ω∈
Ω:
X
(
ω
)=
x
;
Y
(
ω
)=
y
