"
#
$
$
$
$
%
’
’
’
’
"
#
$
$
$
$
"
#
$
$
$
$
%
’
’
’
’
(
)
*
*
*
*
+
,




1.010 Fall 2008
Homework Set #10
Due December 4, 2008 (in class)
1.
Random variables
X
1
, …,
X
50
are independent with the following distributions:
X
1
, …,
X
10
∼
U[0, 1]
X
11
, …,
X
20
∼
N(1, 0.1)
X
21
, …,
X
30
∼
LN(3, 0.2)
X
31
, …,
X
40
∼
Ex[0.4]
X
41
, …,
X
50
∼
Ga(2, 0.2)
where U[0, 1] is the uniform distribution in [0,1], N(
m
,
σ
2
) and LN(
m
,
σ
2
) are the normal
and lognormal distributions with mean value
m
and variance
σ
2
, Ex[
m
] is the exponential
distribution with mean value
m
, and Ga(
n
,
m
) is the gamma distribution that results from
adding
n
iid exponential variables with distribution Ex[
m
]. Find in approximation the
50
probability that
Y
=
"
X
i
exceeds 56.