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STAT 333: Assignment 1
Due
: October 9th in class, or in Pengfei or Martin’s oﬃce no later than 5pm on October 10th.
Note
: Please use the cover page for the assignment.1. A collection of
n
coins is ﬂipped. The outcomes are independent and the
i
th coin comesup heads with probability
α
i
,
i
= 1
,...,n
. Suppose
α
1
= 1
/
2. Find the probabilitythat the total number of heads to appear in the
n
coins will be an even number.2. Suppose
n
letters are randomly selected with replacement from the set
S
=
{
A,B,C,...I
}
(9 letters in total). Let
X
denote the number of distinct letters which appear in the
n
picks. For example, if
n
= 5 and the selected 5 letters are “ACEGA”, then
X
= 4.Calculate
E
(
X
) and Var(
X
).3. A sequence of independent digits (“0” or “1”) is generated, where for each digit theprobability of generating a “1” is
p
, 0
< p <
1. Let
T
be the waiting time (number of steps in the sequence) until we ﬁrst observe “010”.(a) Find
E
(
T
).(b) Find the value of
p
where
E
(
T
) is minimized.
Note:
You may use a computer to calculate this value.4. A mouse is at the center of a maze. Three doors lead out of the maze. Door 1 leadsback to the center after 7 minutes of scampering. Door 2 leads back to the center after6 minutes of scampering. Door 3, however, after 4 minutes of scampering, splits intotwo tunnels, Tunnel
A
and Tunnel
B
. If the mouse chooses Tunnel
A
, it gets out of the maze after 1 minutes. If the mouse chooses Tunnel
B
, it gets out of the maze after3 minutes. This information is displayed in the diagram below.Door 1 Tunnel
A
1 min
❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑
Center
7 mins
❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑
6 mins
4 mins
Door 3
ExitDoor 2 Tunnel
B
3 mins
1
(a) Let
X
be number of minutes for the mouse to get out of the maze. Suppose themouse always chooses at random between any door (or tunnel) at each opportu-nity. Find
E
(
X
) and Var(
X
).(b) Suppose that the mouse always takes a rest before choosing a door (but not thetunnel) – even for the ﬁrst door it picks. Assume that each rest time beforechoosing a door is Uniformly distributed between 0 and 2 minutes, independentlyof anything else. Let
Y
be the total number of minutes that the mouse spentresting before coming out of the maze. Find
E
(
Y
) and Var(
Y
).5. Let
X
1
,...,X
n
be independent Exponential random variables each with rate
λ
i
,
i
=1
,...,n
. (Note that
X
1
,...,X
n
are not necessarily identically distributed.)(a) Find the probability density function of
Y
= min
{
X
1
,X
2
,...,X
n
}
. What type of distribution is the random variable
Y
?(b) Find
P
(
X
1
< X
2
) or equivalently
P
(
X
1
= min
{
X
1
,X
2
}
).(c) Find
P
(
X
1
<
min
{
X
2
,...,X
n
}
) or equivalently
P
(
X
1
= min
{
X
1
,X
2
,...,X
n
}
).
Hint:
Use the results in Part (a) and Part (b).6. Suppose
X
1
,X
2
,...
are independent and identically distributed Uniform random vari-ables over [0
,
1]. Assume that
N
∼
Geo(1
/
2) independently of
X
1
,X
2
,...
.Let
Y
=
N i
=1
X
i
.(a) Find
E
(
Y
).(b) Find
E
(
Y
2
).(c) Find Var(
Y
).2

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