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10/11
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1301 Probability & Statistics I
Assignment 5
Due Date: December 3, 2010
(Hand in your solutions for Questions 1, 5, 18, 23, 27, 34, 37, 38)
1. Let
and
Y
be a continuous random variables distributed in
(
with pdf
(
1
,
0
~
U
X
) )
1
,
0
( )
y
f
Y
and cdf
. Suppose
X
and
Y
are independent. Denote
as an indicator variable such
that
if A
occurs and
otherwise.
()
y
F
Y
1
=
A
I
A
I
0
=
A
I
(a) Show that, for
,
1
0
<
<
t
t
y
t
y
I
I
y
t
y
Y
Y
t
X
P
<
≥
+
=
⎟
⎠
⎞
⎜
⎝
⎛
=
≤
.
(b) Using the result in part (a), show that the cdf of
XY
W
=
is given by
() ()
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≥
<
<
+
≤
=
∫
1
1
1
0
0
0
1
t
t
dy
y
f
y
t
t
F
t
t
F
t
Y
Y
W
.
(c) Using variable substitution and integral by parts for the integral in the expression of
in part (b), show that
t
F
W
( )
t
F
W
can be also expressed as
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≥
<
<
⎟
⎠
⎞
⎜
⎝
⎛
+
≤
=
∫
1
1
1
0
0
0
1
t
t
dx
x
t
F
t
t
t
F
t
Y
W
.
(d) Derive the result in part (c) by first showing that
t
x
t
x
Y
I
I
x
t
F
x
X
X
t
Y
P
<
≥
+
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
=
≤

f
o
r
1
0
<
<
t
.
(e) Find the cdf of
if the cdf of
Y
is given by
XY
W
=
⎪
⎩
⎪
⎨
⎧
≥
<
<
≤
=
1
1
1
0
0
0
2
y
y
y
y
y
F
Y
.
P. 1
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2. Let
, and define
Y
to be the integer part of
()
1
~
Exp
X
1
+
X
.
(a) Find the pdf of
Y
. What wellknown distribution does
Y
have?
(b) Find the conditional distribution of
4
−
X
given
.
5
≥
Y
3. Suppose
( )
p
n
b
p
X
,
~

,
( )
β
α
,
~
Beta
p
.
(a) Find the marginal pdf of
X
. (
X
is said to have a betabinomial distribution.)
(b) Find the condition distribution of
p
, given that
x
X
=
.
4.
The joint density function of
X
and
Y
is given by
(
)
⎩
⎨
⎧
>
>
=
+
−
otherwise
0
0
,
0
if
,
1
y
x
xe
y
x
f
y
x
.
(a) Find the conditional density of
X
,given
y
Y
=
, and that of
Y
, given
.
x
X
=
(b) Find the density function of
XY
Z
=
.
5. An insurance company supposes that each person has an accident parameter and that the
yearly number of accidents of someone whose accident parameter is
λ
is Poisson distributed
with mean
. They also suppose that the parameter value of a newly insured person can be
assumed to be the value of a gamma random variable with parameter
and
. If a newly
insured person has
n
accidents in her first year, find the conditional pdf of her accident
parameter. Also, determine the expected number of accidents that she will have in the
following year.
6. The number of customers using the automatic teller machine in a particular day follows a
Poisson distribution with
180
=
. The amount of money withdrawn by each customer is a
random variable with mean $300 and standard deviation $500. (A negative withdrawal
means that money was deposited.) Find the mean and variance of the total daily withdrawal.
7. Type
i
light bulbs function for a random amount of time having mean
i
μ
and standard
deviation
i
σ
,
. A light bulb randomly chosen from a bin of bulbs is a type 1 bulb with
probability
p
, and a type 2 blub with probability
2
,
1
=
i
p
−
1.
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