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University of Illinois
Spring 2010
ECE 313:
Problem Set 5: Solutions
Conditional Probability, Law of Total Probability, Bayes’ Formula
1.
[Deﬁnition of Conditional Probability]
P
(red apple) =
P
(red)
P
(apple

red) = 0
.
15
P
(apple) =
P
(red apple)
P
(red

apple)
= 0
.
25
2.
[Conditional Probability and Total Probability]
(a) 1
(b) 0
(c)
P
(
BCD

E
)
(d)
P
(
AC
)
3.
[Bernoulli Bus Lines]
(a)
i. What is the probability that you will wait exactly
k
minutes for your bus?
Thirdorder waiting time is a negative binomial random variable,
p
NB
3
(
k
) =
±
k

1
2
²
P
3
(1

P
)
k

3
ii. What is your expected waiting time?
E
[
NB
3] =
3
P
iii. What is the variance of your waiting time?
Var(
NB
3) = 3
±
1

P
P
2
²
(b) The waiting time in this part is ﬁve minutes less than the waiting time in the previous part, i.e.,
Y
=
X

5, therefore
p
Y
(
m
) =
p
NB
3
(
m
+ 5) =
±
m
+ 4
2
²
P
3
(1

P
)
m
+2
(c) Since the Turquoise bus has come and gone, you only have to wait for two buses to arrive. The
waiting time for the second arrival in a Bernoulli process is a secondorder negative binomial:
p
NB
2
(
m
) =
±
m

1
2
²
P
2
(1

P
)
m

2
The fact that the Turquoise bus arrived at 5:03 is irrelevant; the only thing that matters is that
it has already arrived.
(d)
i. What is the probability that you will wait
m
minutes for your bus to arrive?
With probability 0.25, you must wait for two buses to arrive; with probability 0.75, you only
need to wait for one bus to arrive. The probability mass function for your waiting time is
therefore
p
X
(
m
) = 0
.
25
p
NB
2
(
m
) + 0
.
75
p
NB
1
(
m
)
p
X
(
m
) = 0
.
25
±
m

1
2
²
P
2
(1

P
)
m

2
+ 0
.
75
P
(1

P
)
m

1
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View Full Documentii. What is your expected waiting time?
E
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 Spring '10
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