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Massachusetts Institute of Technology
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 5
September 23, 2010
1. (a) Derive the expected value rule for functions of random variables
E
[
g
(
X
)] =
∑
x
g
(
x
)
p
X
(
x
).
(b) Derive the property for the mean and variance of a linear function of a random variable
Y
=
aX
+
b
.
E
[
Y
] =
a
E
[
X
] +
b,
var(
Y
) =
a
2
var(
X
)
.
(c) Derive var(
X
) =
E
[
X
2
]

(
E
[
X
])
2
2. A marksman takes 10 shots at a target and has probability 0.2 of hitting the target with each
shot, independently of all other shots. Let
X
be the number of hits.
(a) Calculate and sketch the PMF of
X
.
(b) What is the probability of scoring no hits?
(c) What is the probability of scoring more hits than misses?
(d) Find the expectation and the variance of
X
.
(e) Suppose the marksman has to pay $3 to enter the shooting range and he gets $2 dollars for
each hit. Let
Y
be his pro±t. Find the expectation and the variance of
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 Spring '11
 Proasin
 Computer Science, Electrical Engineering

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