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Solution for 36217
Wanjie Wang
Teacher: Jiashun Jin
September 23, 2009
If there is any question, please contact me at wwang@stat.cmu.edu
1.(10 Points) Show that
X
=3 is the most likely outcome.
The PMF for Binomial(6, 1/2) is
p
X
(
k
) =
P
(
X
=
k
) =
±
6
k
²
(1
/
2)
k
(1

1
/
2)
6

k
=
±
6
k
²
1
64
So,
P
(
X
= 0) = 1
/
64
, P
(
X
= 1) = 6
/
64
, P
(
X
= 2) = 15
/
64
,
P
(
X
= 3) = 20
/
64
, P
(
X
= 4) = 15
/
64
, P
(
X
= 5) = 6
/
64
, P
(
X
= 6) = 1
/
64.
Obviously the
X
=3 is the most likely outcome.
2. (10 Points)Show the monotonicity of
P
(
X
=
k
)
···
The PMF for Poisson distribution is
p
X
(
k
) =
e

λ
λ
k
k
!
, k
= 0
,
1
,
2
,
···
P
(
X
=
k
)
P
(
X
=
k

1)
=
e

λ λ
k
k
!
e

λ
λ
k

1
(
k

1)!
=
λ
k
So, for
k < λ
,
P
(
X
=
k
)
> P
(
X
=
k

1), which means that
P
(
X
=
k
) increases
monotonically as
k
increases. As
k
increases to
k
≥
λ
,
P
(
X
=
k
)
≤
P
(
X
=
k

1), which means that
P
(
X
=
k
) decreases monotonically as
k
decreases. So,
P
(
X
=
k
) reaches its maximum when
k
is the largest integer not exceeding
λ
.
3. Compute the probability that you will win a prize
···
The distribution of number of prize is Binomial(50, 1/100).
(a) (3 Points) at least once,
P
(
X
= 0) = (99
/
100)
50
≈
0
.
6050
P
(
X
≥
1) = 1

P
(
X
= 0)
≈
0
.
3950
(b) (3 Points) exactly once,
P
(
X
= 1) =
±
50
1
²
(1
/
100)(99
/
100)
49
≈
0
.
3056
1
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View Full Document(c) (3 Points) at least twice.
P
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 Fall '09
 Jin
 Binomial

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