Homework 1 – solution
1.
ˆ
H
is a binomial random variable with parameters
n
= 20 and
p
= 0
.
7.
a) Its probability distribution is given by
P
(
ˆ
H
=
i
) =
‡
20
i
·
0
.
7
i
0
.
3
20

i
for
i
= 0
,
1
,...,
20.
b)
E
(
ˆ
H
) =
np
= 14.
c) Std(
ˆ
H
) =
q
np
(1

p
) = 2
.
05.
2. The lifetime of any given bulb is
T
∼
N
(10000
,
1000).
a)
P
(
T >
12000) = 1

F
(12000) = 1

Φ
‡
12000

10000
1000
·
= 1

Φ(2) =
1

0
.
977 = 0
.
023.
b)
P
(
T <
9000) =
F
(9000) = Φ
‡
9000

10000
1000
·
= Φ(

1) = 0
.
159.
c)
P
(replacing at least one bulb) = 1

P
(replacing no bulbs) = 1

P
(all
T
i
>
5000) = 1

P
(
T >
5000)
100
= 1

(1

F
(5000))
100
= 1

(1

Φ(

5))
100
= 0 (up to three decimal points).
d)
P
(all bulbs burining out) =
P
(all
T
i
<
15000) =
P
(
T <
15000)
100
=
(
F
(15000))
100
= (Φ(5))
100
= 1 (up to three decimal points).
3. Let
X
be the (random) volume to be put in a vial. We know that
X
∼
N
(
μ,σ
)
,
with
σ
= 0
.
1 and
μ
unknown (to be determined).
The requirement that the fraction of underﬁlled vials be 0.01 is expressed
as
P
(
X <
2
.
5) = 0
.
01
.
Standardizing, we obtain
P
±
Z <
2
.
5

μ
σ
¶
= 0
.
01
,
from which it follows that
2
.
5

μ
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 Spring '08
 Perevalov
 Probability distribution, Probability theory, Cumulative distribution function, Discrete probability distribution, dx

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