3? I accepted
both interpretations.
Interpretation:
N
is a real number.
The occurrence of errors is governed by a
Poisson arrival process. We only notice 90% of errors and, assuming that this is
independent for each error, the occurrence of errors we
find
is a Poisson arrival
process with rate 0
.
9
·
1
/
4. So
N
is exponentially distributed with rate 0
.
9
/
4
and
E
(
N
) =
1
0
.
9
/
4
=
4
0
.
9
≈
4
.
44
.
Interpretation:
N
is an integer.
The number of misprints we
find
per page is
given by a Poisson distribution with rate 0
.
9
/
4, so the probability that we find
at least one misprint on a single page is 1

e

0
.
9
/
4
. Therefore,
N
has geometric
distribution with
p
= 1

e

0
.
9
/
4
and
E
(
N
) =
1
1

e

0
.
9
/
4
≈
4
.
96
.
(c) [6 points] What is the probability that at least 2 of the first 50 pages contain at
least 3 misprints each?
Solution:
Using the Poisson distribution, the probability that a single page
contains at least three misprints is
p
= 1

P
(none)

P
(one)

P
(two)
= 1

e

1
/
4
(1
/
4)
0
0!

e

1
/
4
(1
/
4)
1
1!

e

1
/
4
(1
/
4)
2
2!
≈
0
.
00216
.
Using the binomial distribution, the probability that at least two of the first 50
pages contain at least one misprint is
1

P
(none)

P
(one) = 1

(1

p
)
50

50
1
(1

p
)
49
p
≈
0
.
534%
.
3. The type of lightbulb used in your refrigerator has a lifetime that is exponentially dis
tributed with a mean of 2 years. You replace the bulb every time it fails.
(a) [6 points] What is the probability that the first lightbulb lasts less than a year?
Page 2
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Math 431: Exam 2
Solution:
The c.d.f. for the exponential distribution with rate 1
/
2 is
F
(
t
) =
1

e

t/
2
.
So the probability that first lightbulb dies within the first year is
1

e

1
/
2
.
(b) [7 points] What is the probability that the the first three lightbulbs last at least 5
years (all together)?
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '12
 Miller
 Normal Distribution, Probability, Probability theory, Exponential distribution

Click to edit the document details