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PROBABILITY AND DECISION MAKING
45730
QUIZ 2 (OFFLINE)
Problem 1
(
Expectation and Variance
)
.
You and your friend play the following game. You toss
a six sided dice. Your score, denoted by X, is the number that appears on the dice and your friend’s
score, denoted by Y is 6 minus the number that appears on the dice. The person who gets the higher
score wins the game. In case both scores are equal, the game results in a tie.
(1.1) What is your expected score E[X]?
Solution.
E
[
X
] =
∑
6
i
=1
1
6
·
i
= 3
.
5
±
(1.2) What is the expected score of your friend E[Y]?
Solution.
Y
= 6

X
. Therefore,
E
[
Y
] =
E
[6

X
] = 6

E
[
X
] = 2
.
5 (by linearity of expectation).
±
(1.3) What is Var[X]?
Solution.
V ar
[
X
] =
∑
6
i
=1
1
6
·
(
i

E
[
X
])
2
= 2
.
92.
±
(1.4) What is Var[Y]?
Solution.
V ar
[
Y
] =
V ar
[6

X
] =
V ar
[
X
] = 2
.
92.
±
(1.5) Let
Z
be a random variable such that
Z
=
X

Y
. What is E[Z]?
Solution.
Using linearity of expectation,
E
[
Z
] =
E
[
X
]

E
[
Y
] = 1.
±
(1.6) What is the value of Var[Z]?
Solution.
Note that
Z
=
x

(6

X
) = 2
X

6. Therefore,
V ar
[
Z
] = 4
·
V ar
[
X
] = 11
.
67.
±
(1.7) What is the probability that Z is greater than zero? Note that this is the probability that you
win the game.
Solution.
Z >
0
⇒
(2
X

6)
>
0
⇒
X >
3. Thus,
P
(
Z >
0) =
1
2
.
±
(1.8) Now, suppose you play the following modiﬁed game. You and your friend both toss a six sided
dice independently and let
X
denote the number that you get and
Y
denote the number that
your friend gets. Again, let
Z
=
X

Y
. What is the value of E[Z] and Var[Z]?
Solution.
From part 1, we know
E
[
X
] = 3
.
5. Since,
X
are
Y
are i.i.d,
E
[
X
] =
E
[
Y
] = 3
.
5.
Therefore,
E
[
Z
] = 0 and
V ar
[
Z
] =
V ar
[
X
] +
V ar
[
Y
] = 5
.
83.
±
Problem 2.
You and your friend play the following game. You toss a six sided dice and the number
that appears on the dice is your score, denoted by X. Your friend tosses a fair coin six times and his
score is the number of times the toss resulted in a heads, denoted by Y. The person getting a higher
score wins. In case both scores are equal, the game results in a tie.
(2.1) What is the probability that the game will result in a tie, i.e.
P
(
X
=
Y
)?
1
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QUIZ 2 (OFFLINE)
Solution.
For
i
= 1
, . . . ,
6,
P
(
X
=
i, Y
=
i
) =
1
6
·
C
6
i
2
6
.
Note that
C
6
0
= 1,
C
6
1
= 6,
C
6
2
= 15,
C
6
3
= 20,
C
6
4
= 15,
C
6
5
= 6,
C
6
6
= 1. Thus,
P
(game results in a tie)
=
6
X
i
=1
P
(
X
=
i, Y
=
i
) =
1
6
6
X
i
=1
C
6
i
2
6
=
1
6
·
6 + 15 + 20 + 15 + 6 + 1
64
=
1
6
·
63
64
.
±
(2.2) What is the probability that you win the game, i.e.
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 Spring '08
 ravi
 Decision Making

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