UCSD ECE153
Handout #33
Prof. Young-Han Kim
Tuesday, June 1, 2010
Final Examination (Fall 2008)
1.
Order statistics.
Let
X
1
,X
2
,X
3
be independent and uniformly drawn from the interval [0
,
1]. Let
Y
1
be
the smallest of
X
1
,X
2
,X
3
, let
Y
2
be the median (second smallest) of
X
1
,X
2
,X
3
, and
let
Y
3
be the largest of
X
1
,X
2
,X
3
. For example, if
X
1
=
.
3
,X
2
=
.
1
,X
3
=
.
7, then
Y
1
=
.
1
,Y
2
=
.
3
,Y
3
=
.
7. The random variables
Y
1
,Y
2
,Y
3
are called the
order statistics
of
X
1
,X
2
,X
3
.
(a) What is the probability P
{
X
1
≤
X
2
≤
X
3
}
?
(b) Find the pdf of
Y
1
.
(c) Find the pdf of
Y
3
.
(d) (Di±cult.) Find the pdf of
Y
2
.
(Hint:
Y
2
≤
y
if and only if at least two among
X
1
,X
2
,X
3
are
≤
y
.)
2.
Fair coins.
We are given two coins: Coin 1 with bias (=probability of heads) 1
/
2 and Coin 2
with random bias
P
∼
Unif[0
,
1]. We pick one at random and ²ip it three times
independently. The value of the bias does not change during the sequence of tosses.