University of Illinois
Spring 2010
ECE 313:
Problem Set 10
Due:
Wednesday, April 7 at 4 p.m.
Reading:
Ross, Chapter 5; Lecture Notes 2628.
Noncredit Exercises:
DO NOT turn these in.
Chapter 5:
Problems 11,15 and 31;
Theoretical Exercises 18,30.
This Problem Set contains seven problems.
1.
[Moments of random variables]
Theoretical Exercises, Chapter 5 of the textbook, Problem 5.5.
2.
[Gaussian Random Variables]
Let
X
be a Gaussian random variable with mean
μ
=

1 and variance
σ
2
= 4.
(a) Find the mean and variance of 2
X
+ 5.
Let Φ(
x
) denote the CDF of a standard Gaussian random variable, and let
Q
(
x
) =
1

Φ(
x
). Suppose that Calculator A can evaluate only Φ(
x
) and only for nonnegative
values of
x
. On the other hand, suppose that Calculator B can evaluate only
Q
(
x
),
again only for
x
≥
0. Both calculators can perform standard functions, like addition
and multiplication. For each of the probabilities in parts (
b
) through (
e
), write down
two
alternative expressions: one for evaluation using Calculator A, and the other for
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 Spring '08
 Milenkovic,O
 Normal Distribution, Probability theory, Cumulative distribution function, CDF, Gaussian random variable

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