Statistics 851 (Fall 2013)
September 27, 2013
Prof. Michael Kozdron
Lecture #11: Continuity of Probability (continued)
We will now apply the continuity of probability theorem to prove that the function
F
(
x
) =
P
{
(
−∞
, x
]
}
,
x
∈
R
, defined last lecture is actually a distribution function.
Theorem 11.1.
Consider the real numbers
R
with the Borel
σ
algebra
B
, and let
P
be a
probability on
(
R
,
B
)
. The function
F
:
R
→
[0
,
1]
defined by
F
(
x
) =
P
{
(
−∞
, x
]
}
for
x
∈
R
is a distribution function; that is,
(i)
lim
x
→−∞
F
(
x
) = 0
and
lim
x
→∞
F
(
x
) = 1
,
(ii)
F
is rightcontinuous, and
(iii)
F
is increasing.
Proof.
In order to prove that
F
(
x
) =
P
{
(
−∞
, x
]
}
is a distribution function we need to
verify that the three conditions in definition are met. We will begin by showing (ii). Thus,
to show that
F
is rightcontinuous, we must show that if
x
n
is a sequence of real numbers
converging to
x
from the right, i.e.,
x
n
↓
x
or
x
n
→
x
+, then
F
(
x
n
) converges to
F
(
x
), i.e.,
lim
x
n
→
x
+
F
(
x
n
) =
F
(
x
)
.
However, this follows immediately from the continuity of probability theorem (actually it
follows directly from Exercise 10.3 which follows directly from Theorem 10.2) by noting that
if
x
n
→
x
+, then
(
−∞
, x
1
]
⊇
(
−∞
, x
2
]
⊇
· · ·
⊇
(
−∞
, x
j
]
⊇
(
−∞
, x
j
+1
]
⊇
· · ·
⊇
(
−∞
, x
]
and
∞
n
=1
(
−∞
, x
n
] = (
−∞
, x
]
so that
lim
x
n
→
x
+
F
(
x
n
) = lim
n
→∞
P
{
(
−∞
, x
n
]
}
=
P
∞
n
=1
(
−∞
, x
n
]
=
P
{
(
−∞
, x
]
}
=
F
(
x
)
.
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 Fall '14
 Statistics, Probability, Probability theory, Probability space, lim F

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