ECE 534
Fall 2009
September 16, 2009
Probability Quiz Solutions
1. (8 pts)
Warmup
(a) Consider the probability space (Ω
,
F
, P
). If
A
∈ F
is such that
P
(
A
) = 1, then prove that
for any
C
∈ F
,
P
(
A
∩
C
) =
P
(
C
). (Note: you cannot assume that
A
= Ω.)
Ans:
P
(
A
∩
C
) =
P
(
A
) +
P
(
C
)

P
(
A
∪
C
). But 1
≥
P
(
A
∪
C
)
≥
P
(
A
) = 1. Thus
P
(
A
∪
C
) = 1,
which implies that
P
(
A
∩
C
) = 1 +
P
(
C
)

1 =
P
(
C
).
(b) Determine if the following statement is True or False:
Every (measurable) function of
continuoustype random variable is also a continuoustype random variable.
If you say
“True”, explain why; if you say “False” provide a counterexample.
Ans:
False. We saw a counterexample in class. Hardlimiting a
N
(0
,
1) random variable to the interval
[0
,
1] produces a mixed random variable with point masses at 0 and 1.
(c) How many rolls of a fair die will it take on average to see a 3?
Explain your reasoning
carefully.
Ans:
It is easy to see that the number of rolls it will take to see a 3 is a Geom(
ρ
) random variable
with
ρ
= 1
/
6. Thus the average number of rolls is 6. Another way to see this it to realize that if
m
is the average, then
m
has to satisfy
m
= 1 +
m
(5
/
6), since if the first roll is not a 3, which happens
with probability 5/6, the average from second roll onwards will be
m
as well.
2. (6 pts)
Faulty Memory.
A binary memory reader is faulty in a way that a bit ‘0’ is read as a ‘1’
with probability 0.2, and a bit ‘1’ is read as a ‘0’ with probability 0.1. Assume that the bits in
the memory are equally likely to be ‘0’ or ‘1’. Also assume that the errors made by the reader are
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 Spring '09
 all
 Probability, Probability theory, V. Veeravalli

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