Part 1 – Random Variables and Distributional Quantities
1.
The distribution function (df) or cumulative distribution function (cdf) and
survival function (sf) of a random variable (rv)
X
are deﬁned by
F
(
x
) =
Pr
{
X
≤
x
}
and
S
(
x
) =
Pr
{
X > x
}
= 1

F
(
x
)
for all
x.
(a) The df
F
(
x
) and sf
S
(
x
) satisfy the following four conditions:
(1) 0
≤
F
(
x
)
≤
1 for all
x
(0
≤
S
(
x
)
≤
1 for all
x
).
(2)
F
(
x
) is nondecreasing (
S
(
x
) is nonincreasing).
(3)
F
(
x
) is rightcontinuous (
S
(
x
) is rightcontinuous).
(4)
F
(
∞
) = 0 and
F
(
∞
) = 1 (
S
(
∞
) = 1 and
S
(
∞
) = 0).
(b) Any function satisfying the above four conditions is a df (sf) of some random
variable.
(c)
Pr
{
a < X
≤
b
}
=
F
(
b
)

F
(
a
) =
S
(
a
)

S
(
b
).
(d)
Pr
{
X
=
a
}
=
Pr
{
X
≤
a
} 
Pr
{
X < a
}
=
F
(
a
)

F
(
a

).
(e) If
F
(
x
) is continuous at
x
=
a
, then
Pr
{
X
=
a
}
= 0. If
F
(
x
) is not continuous
at
x
=
a
, then
Pr
{
X
=
a
}
=
F
(
a
)

F
(
a

) is the jump size of
F
(
x
) at
x
=
a
.
2.
Three types of random variables:
(a) Continuous:
X
takes all the values in some interval such as (
a, b
), [0
,
∞
), (
∞
,
∞
),
and so on.
(b) Discrete:
X
takes only ﬁnite values
{
x
1
, ..., x
n
}
or countable values
{
x
1
, x
2
, ...
}
.
(c) Mixed:
X
takes all the values in some interval and some ﬁnite or countable values.
3.
For a continuous rv
X
with df
F
(
x
):
(a)
Pr
{
X
=
a
}
= 0 for all
a
.
(b)
Pr
{
a < X
≤
b
}
=
Pr
{
a
≤
X
≤
b
}
=
Pr
{
a < X < b
}
=
Pr
{
a
≤
X < b
}
=
F
(
b
)

F
(
a
)
.
4.
The density function or probability density function (pdf)
of a continuous rv
X
is given by
f
(
x
) =
F
0
(
x
) =

S
0
(
x
) with
R
∞
∞
f
(
y
)
dy
= 1
,
F
(
x
) =
Z
x
∞
f
(
y
)
dy
and
S
(
x
) =
Z
∞
x
f
(
y
)
dy.
In addition,
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 Fall '09
 david
 Probability theory, probability density function, Cumulative distribution function, Probability mass function, xj

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