This preview shows pages 1–3. Sign up to view the full content.
Homework 3
•
Oncampus students:
Due by 10:40 AM on Monday, Oct. 3rd, 2011
•
Offcampus students:
Due by 11:55 PM on Monday, Oct. 10th, 2011
(As usual, the problems labeled “SS” are for selfstudy and should not be submitted for grading.)
B
INOMIAL
, G
EOMETRIC
,
AND
P
OISSON
P
ROBABILITIES
SS1. On a very noisy communication channel, the probability of a packet being received correctly
is 0.2. If the packet is not received correctly, then the destination does not send an acknowledgment
(ACK). The source repeatedly transmits the packet until an ACK is received.
(a) What is the probability that three transmissions are required? [0.128]
(b) What is the probability that more than three transmissions are required? [0.512]
(c) Given that the ﬁrst three transmissions were failures, what is the probability that more
than three more transmissions are required? [0.512]
(d) Find a formula for the probability that
Δ
more transmissions are required given that the
ﬁrst
j
transmissions were failures. [
(
1

p
)
Δ

1
p
]
(e) Explain why geometric probabilities are said to be “memoryless”.
1. In 2003, there were many media reports about the number of shark attacks in Florida. At
the end of the year, there were a total of 30 unprovoked shark attacks. By comparison, there
were 246 shark attacks over the prior ten years.
(a) Give an expression for the probability of
more than thirty
shark attacks occurring in
a year based on the historical data (don’t include the data for 2003). Use a Gaussian
approximation to give an approximate numerical answer.
(b) Suppose that the probability that there are more than 30 shark attacks in a given year is
0.1. Find the probability that there is
at least one
year with more than 30 shark attacks
in 10 year period.
(c) Use the historical data to answer this question. Shark attacks occur primarily during
warm weather, so suppose that shark attacks only occur over a 30 week period. In
2003, the media made a big deal about two shark attacks during the same week. What
is the probability of having
at least two
shark attacks during a given week?
(d) Using the assumptions of the previous part of this problem, what is the probability of
having at least one week with
three
or more shark attacks during a given year?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2. (Modiﬁed from
Stark and Woods
, 3rd ed.)
(Independence of events in disjoint arrivals for Poisson law) The average number of cars
arriving at a tollbooth per minute is
λ
and the probability of
k
cars in the interval
(
0
,
T
)
is
P
(
k
;0
,
T
) =
e

λ
T
[
λ
T
]
k
k
!
.
Consider two disjoint, that is nonoverlapping intervals, say
(
0
,
t
1
)
and
(
t
1
,
T
)
. Then for the
Poisson law:
P
[
n
1
cars in
(
0
,
t
1
)
and
n
2
cars in
(
t
1
,
T
)] =
P
[
n
1
cars in
(
0
,
t
1
)]
P
[
n
2
cars in
(
t
1
,
T
)]
Thus events in disjoint intervals are independent.
(a) Show that
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '11
 Wang

Click to edit the document details