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UCLA Electrical Engineering Department
EE132B
HW Set #2
Professor Izhak Rubin
Problem 1
For the Gaussian distribution with mean
μ
and variance
2
σ
, find the moment generating function.
Using the moment generating function, calculate the mean and the variance.
Hint: The probability density function for a Gaussian random variable x with mean
and variance
2
is given by
(
29
(
29
2
2
1
2
2
1
2
x
X
f
x
e
πσ


=
.
Problem 2
Consider the following probability density function:
(
29
2
x
X
e
f
x
λ

=
for
(
29
,
x
∈ ∞ ∞
.
1)
Find the mean directly.
2)
Find the variance directly.
3)
Find the moment generating function.
4)
Find the mean and the variance from the moment generating function.
Problem 3
A coin is flipped until heads occur twice. Define two random variables X and Y to be the trial
numbers at which the first and the second heads are observed. Assume that at any trial, the
probability that a head occurs is
(
29
0,1
p
∈
.
1) Show that the joint probability is given by
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 Spring '09
 IzhakRubin
 Electrical Engineering

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