1 Random Variables
In applications we are interested in quantitative properties of experimental results.
Example:
Toss a coin three times and count the number of heads. The sample space is
S
=
{
(
t,t,t
)
,
(
t,t,h
)
,
(
t,h,t
)
,
(
h,t,t
)
,
(
t,h,h
)
,
(
h,t,h
)
,
(
h,h,t
)
,
(
h,h,h
)
}
.
The random variable
X
counts the number of heads. Thus if
w
= (
t,t,h
) occurs then
X
(
t,t,h
) = 1
.
Assigning probabilities to value of random variables is done by connecting probability associates with
outcomes of an experiment with the values of the random variables.
Example:
(continued) Assume all elements in
S
are equally likely. Then
X
= 1 corresponds to the set
{
(
t,t,h
)
,
(
t,h,t
)
,
(
h,t,t
)
}
thus
P
(
X
= 1) =
P
(
{
(
t,t,h
)
,
(
t,h,t
)
,
(
h,t,t
)
}
) = 3
/
8
.
Let
B
=
{
2
,
3
}
by
X
∈
B
we mean the subset
A
=
{
(
t,h,h
)
,
(
h,t,h
)
,
(
h,h,t
)
,
(
h,h,h
)
}
of
S
such that
X
(
w
)
∈
B
for all
w
∈
A
so
P
(
X
∈
B
) =
P
(
A
) =
4
8
= 0
.
5
.
By
X
=
j
we mean the set
A
=
{
w
∈
S
:
X
(
w
) =
j
}
.
So
X
= 0 is the set
A
=
{
(
t,t,t
)
}
and
P
(
X
= 0) =
P
(
A
) = 1
/
8
,
etc.
Deﬁnition:
A function
X
:
S
→ <
mapping elements of the sample space into the real numbers will be
called a
random variable
.
Remark: A random variable is a deterministic function. What is random is the selection of
w
∈
S
.
Example:
Suppose you select a point at random in the unit circle
{
(
x,y
) :
x
2
+
y
2
≤
1
}
. Let
D
be the
distance of the selected point to the origin
p
x
2
+
y
2
. Then
D
is a random variable.
1.1 Discrete Random Variables
Deﬁnition:
Discrete Random Variables: If the possible values of
X
are ﬁnite or countable we say that
X
is a discrete random variable.
Deﬁnition:
Let
x
1
,x
2
,...
denote the possible values of
X
then
p
(
x
i
) =
P
(
X
=
x
i
) =
P
(
w
∈
S
:
X
(
w
) =
x
i
)
is called the probability mass function. Notice that
p
(
x
i
)
≥
0 and that
∞
X
i
=1
p
(
x
i
) =
X
i
P
(
w
∈
S
:
X
(
w
) =
x
i
) =
P
(
S
) = 1
.
Convenient notation (not used in book):
p
X
(
a
) =
P
(
X
=
a
). This notation is useful when you want to
emphasize that you are refereing to random variable
X
. When there is no danger of confusion we drop the
subscript.
Example:
(continued)
p
(0) = 1
/
8,
p
(1) = 3
/
8
, p
(2) = 3
/
8,
p
(3) = 1
/
8.
The probability mass function summarizes all probability information associated with the random vari
able.
1
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View Full DocumentExample:
(continued)
P
(
X
≤
1) =
p
(0) +
p
(1) = 0
.
5
. P
(1
≤
X
≤
2) =
p
(1) +
p
(2) = 3
/
4,
P
(
X >
1) =
p
(2) +
p
(3) = 0
.
5.
Although the probability mass function contains all information w.r.t. a discrete random variable, the
cumulative distribution function
is frequently used.
Deﬁnition:
The cumulative distribution function (cdf)
F
of a random variable
X
is given by
F
X
(
a
) =
P
(
X
≤
a
)
.
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 Spring '04
 GALLEGO
 Probability theory, probability density function

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