HW#7, Solutions
AMS 311 Probability Theory (Spring 2011)
Part 1 (10 points):
PRINT
your name and StonyBrook ID (
Last, First, #ID
) on the top on your work
sheet. Staple your HW if more than 1 pages.
Part 2: Problems (90 points)
Write down your work step by step to get full score.
(1) (30 points)
(a) Denote p
i
=P(the rat dies by poison  the rat starts in Cell
i
). Then conditioning on the first
door chosen, we get:
p
1
=(1/5)*0+(2/5)*1+(2/5)p
3
p
2
=1*p
3
p
3
=(1/5)*1+(2/5)*p
1
+(2/5)*p
2
Solve, getting p
1
=8/11, p
2
=p
3
=9/11.
(b) Let Y be the number of minutes the rat lives in total. Let t
i
= E(Y  the rat starts in Cell i). We
want to compute t
1
. Note that the rat wanders in Cell i for an expected amount of time (
i+i
2
)/2
minutes, since the expected value of a Uniform(
i, i
2
) is (
i+i
2
)/2. Then, conditioning on the first
door chosen, we get:
t
1
=(1+1
2
)/2+(1/5)*0+(2/5)*0+(2/5)*t
3
t
2
=(2+2
2
)/2+1*t
3
t
3
=(3+3
2
)/2+(1/5)*0+(2/5)*t
1
+(2/5)*t
2
Solve, we get t
1
=87/11 mins
(c) Let q
i
= P( the rat visits Cell 2 before he dies  the rat starts in Cell i). Then, conditioning on
the first door chosen,
we get:
q
1
=(1/5)*0+(2/5)*0+(2/5)*q
3
q
2
=1
q
3
=(1/5)*0+(2/5)*q
1
+(2/5)*q
2
Solve, we get q
1
=4/21
(2). (30 points)
Homework is due
at the beginning of class
on its due date. You must be
in class
and
on time
to submit
your homework. No homework will be accepted via email. No late homework will be accepted. Your
lowest homework grade will be dropped before computing your average.
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 Spring '08
 Tucker,A
 Probability theory, probability density function, lowest homework grade

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