EE 178
Handout #4
Probabilistic Systems Analysis
January 24, 2008
Homework #3
Due Thursday, January 31
1.
Liars.
Of the 100 people in a village, 50 always tell the truth, 30 always lie, and 20 always
refuse to answer. A sample of 30 people are drawn with replacement.
a. What is the probability that the sample will contain 10 people from each category?
b. What is the probability that the sample will have exactly 12 liars?
2.
Binary vectors.
The sample space Ω of an experiment consists of all 8dimensional binary
vectors, e.g., every member of Ω has the form
ω
= (
ω
0
, . . . , ω
7
), where
ω
i
is 0 or 1. The
probability law
P
assigns a probability of 1
/
2
8
to each of the 2
8
elements in Ω.
Find the pmfs describing the following random variables:
a.
W
(
ω
) =
∑
7
i
=0
ω
i
, i.e., the number of 1’s in the binary vector.
b.
X
(
ω
) = 1 if there are an even number of 1’s in
ω
and 0 otherwise.
c.
Y
(
ω
) =
ω
4
, i.e., the value of the 4th coordinate of
ω
.
d.
Z
(
ω
) = max
i
(
ω
i
).
3.
Heads and tails.
Consider a sequence of five independent fair coin tosses. Define random
variable
X
to be the number of times that heads is followed immediately by a tail. Find
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 Spring '08
 Hewlett
 Probability, Probability theory, Sinéad

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