Utah State University
ECE 6010
Stochastic Processes
Homework # 2 Solutions
1. Suppose
X
is a r.v. with c.d.f.
F
X
. Prove the following:
(a)
F
X
is nondecreasing.
Let
b > a
.
F
X
(
b
)

F
X
(
a
) =
P
(
X
≤
b
)

P
(
X
≤
a
) =
P
(
X
≤
a
) +
P
(
a < X
≤
b
)

P
(
X
≤
a
)
=
P
(
a < X
≤
b
)
≥
0
.
So
F
X
(
b
)
≥
F
X
(
a
) for
b > a
, which means
F
X
is nondecreasing.
(b) lim
a
→∞
F
X
(
a
) = 1.
lim
a
→∞
F
X
(
a
) = lim
a
→∞
P
(
X
≤
a
) =
P
(
{
ω
:
X
(
ω
)
≤
a
}
) =
P
(Ω) = 1
.
(c) lim
a
→∞
F
X
(
a
) = 0.
lim
a
→∞
F
X
(
a
) =
lim
a
→∞
P
(
{
ω
:
X
(
ω
)
≤
a
}
) =
P
(
∅
) = 1
.
(d)
F
X
is right continuous.
Let
B
n
=
{
ω
∈
Ω :
X
(
ω
)
≤
a
+ 1
/n
}
for
n
= 1
,
2
, . . .
. Note that this is a nested
sequence,
B
1
⊃
B
2
⊃ · · ·
. We have
lim
n
→∞
F
X
(
a
+ 1
/n
) = lim
n
→∞
P
(
B
n
) =
P
( lim
n
→∞
B
n
)
by continuity of probability. But
lim
n
→∞
B
n
=
{
ω
:
X
(
ω
)
≤
a
}
, so
lim
n
→∞
F
X
(
a
+ 1
/n
) =
P
(
X
≤
a
)
.
Since the limit from the right is equal to the limiting value, we have right conti
nuity.
(e)
P
(
a < X
≤
b
) =
F
X
(
b
)

F
X
(
a
) if
b > a
.
P
(
a < X
≤
b
) =
P
(
X
≤
b
)

P
(
X
≤
a
) =
F
X
(
b
)

F
X
(
a
).
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(f)
P
(
X
=
a
) =
F
X
(
a
)

lim
b
→
a

F
X
(
b
)
.
P
(
X
=
a
) =
P
(
X
≤
a
)

P
(
X < a
) =
F
X
(
a
)

lim
b
→
a

F
X
(
b
).
Also, find expressions for
P
(
a
≤
X
≤
b
),
P
(
a
≤
X < b
) and
P
(
a < X < b
) in terms of
F
X
.
P
(
a
≤
X
≤
b
) =
P
(
a < X
≤
b
) +
P
(
X
=
a
) =
F
X
(
b
)

F
X
(
a
) + (
F
X
(
a
)

lim
b
→
a

F
X
(
b
))
.
P
(
a
≤
X < b
) =
P
(
a < X
≤
b
) +
P
(
X
=
a
)

P
(
X
=
b
)
=
F
X
(
b
)

F
X
(
a
) + (
F
X
(
a
)

lim
c
→
a

F
X
(
c
))

(
F
X
(
b
)

lim
c
→
b

F
X
(
c
))
2. Show that the following are valid p.m.f.s:
(a) Binomial:
f
X
(
a
) =
n
!
/
((
n

a
)!
a
!)
π
a
(1

π
)
n

a
if
a
∈ {
0
,
1
, . . . , n
}
.
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 Spring '08
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