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Homework Set No. 8 – Probability Theory (235A), Fall 2009
Posted: 11/17/09 — Due: 11/24/09
1. (a)
Prove that if
X,
(
X
n
)
∞
n
=1
are random variables such that
X
n
→
X
in probability
then
X
n
=
⇒
X
.
(b)
Prove that if
X
n
=
⇒
c
where
c
∈
R
is a constant, then
X
n
→
c
in probability.
(c)
Prove that if
Z,
(
X
n
)
∞
n
=1
,
(
Y
n
)
∞
n
=1
are random variables such that
X
n
=
⇒
Z
and
X
n

Y
n
→
0 in probability, then
Y
n
=
⇒
Z
.
2. (a)
Let
X,
(
X
n
)
∞
n
=1
be integervalued r.v.’s. Show that
X
n
=
⇒
X
if and only if
P
(
X
n
=
k
)
→
P
(
X
=
k
) for any
k
∈
Z
.
(b)
If
λ >
0 is a ﬁxed number, and for each
n
,
Z
n
is a r.v. with distribution Binomial(
n,λ/n
),
show that
Z
n
=
⇒
Poisson(
λ
)
.
3.
Let
f
(
x
) = (2
π
)

1
/
2
e

x
2
/
2
be the density function of the standard normal distribution,
and let Φ(
x
) =
R
x
∞
f
(
u
)
du
be its c.d.f. Prove the inequalities
1
x
+
x

1
f
(
x
)
≤
1

Φ(
x
)
≤
1
x
f
(
x
)
,
(
x >
0)
.
(1)
Note that for large
x
this gives a very accurate twosided bound for the tail of the normal
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This note was uploaded on 11/13/2011 for the course STAT 235A taught by Professor Danromik during the Fall '11 term at UC Davis.
 Fall '11
 DanRomik
 Probability

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