Stat 116 Homework 3 Solutions
April 16, 2008
1. Let
A
and
B
be the number of people on a given flight from airline A and B respectively. Notice the
probability that any given person checks in is 9/10 so this random variable is distributed Bernoulli
with 9/10 as the parameter.
Therefore since
A
and
B
are the sums of such random variables the
distribution for
A
and
B
must be binomial, i.e.
P
(
A
=
k
) =
10
k
9
10
k
1
10
10

k
,
P
(
B
=
k
) =
20
k
9
10
k
1
10
20

k
Then
P
(A is overbooked) =
P
(
A
= 10) =
9
10
10
.
=
.
3486
P
(B is overbooked) =
P
(
B
≥
19) = 20
9
10
19
1
10
+
9
10
20
.
=
.
3917
So airline B have a greater chance of being oversold.
2. Let
ξ
i
be the number of throws before ith distinct number to appear for the first time after i1 distinct
numbers has appeared.
ξ
i
has a geometric distribution with success probability
p
i
=
6

i
+1
6
, so
E
[
ξ
i
] =
1
p
i
.
ξ
=
6
X
i
=1
ξ
i
E
[
ξ
] =
6
X
i
=1
E
[
ξ
i
]
=
6
X
i
=1
1
6

i
+1
6
= 14
.
7
1
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3. Let the number of signals received be
ξ
.
P
(
η
=
k
) =
X
n
≥
k
P
(
ξ
=
n
)
P
(
η
=
k

ξ
=
n
)
=
X
n
≥
k
λ
n
n
!
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 Spring '07
 Staff
 Bernoulli, Probability, Probability theory, probability density function, approximately Poisson, ξi

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