Content Actions

Download PDF
Save to del.icio.us (?)Add this content to the social bookmarking site del.icio.us, where you can share your favorite links with others.

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

This content is ...

Affiliated with (?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.

This module is included inLens: Rice University OpenCourseWare
By: OpenCourseWare ConsortiumAs a part of collection:"Digital Communication Systems"
Click the "Rice University OCW" link to see all content affiliated with them.
Rice University OCW

Related material

Collections using this content

Digital Communication Systems

Tags: (?)
These tags come from the endorsement, affiliation, and other lenses that include this content.
Technology

Homework 1 of Elec 430

Module by: Behnaam Aazhang

Summary: First homework of Elec 430

Elec 430 homework set 1. Rice University Department of Electrical and Computer Engineering.

Problem 1

The current I

in a semiconductor diode is related to the voltage V

by the relation I=ⅇV-1

I V 1

. If V

is a random variable with density function f V x=12ⅇ-|x|

f V x 1 2 x

for -∞<x<∞

, find fIy

f I y

; the density function of I

Problem 2

2.a)

Show that if AB={}

A B {}

then PrA≤Pr B c

A B c

2.b)

Show that for any A

, B

, C

we have PrA⋃B⋃C=PrA+PrB+PrC-PrA⋂B-PrA⋂C-PrB⋂C+PrA⋂B⋂C

A B C A B C A B A C B C A B C

2.c)

Show that if A

and B

are independent the PrA⋂ B c =PrAPr B c

A B c A B c

which means A

and Bc

are also independent.

Problem 3

Suppose X

is a discrete random variable taking values 012…n

0 1 2 … n

with the following probability mass function pXk=n!k!n-k!θk1-θn-kifk=012…n0otherwise

p X k n k n k θ k 1 θ n k k 0 1 2 … n 0

with parameter θ∈01

θ 0 1

3.a)

Find the characteristic function of X

3.b)

Find X¯

and σ X 2

σ X 2

Hint: See problems 3.14 and 3.15 in Proakis and Salehi

Problem 4

Consider outcomes of a fair dice Ω= ω 1 ω 2 ω 3 ω 4 ω 5 ω 6

Ω ω 1 ω 2 ω 3 ω 4 ω 5 ω 6

. Define events A={ω|an even number appears}

A ω an even number appears ω

and B={ω|a number less than 5 appears}

B ω a number less than 5 appears ω

. Are these events disjoint? Are they independent? (Show your work!)

Problem 5

This is problem 3.5 in Proakis and Salehi.

An information source produces 0 and 1 with probabilities 0.3 and 0.7, respectively. The output of the source is transmitted via a channel that has a probability of error (turning a 1 into a 0 or a 0 into a 1) equal to 0.2.

5.a)

What is the probability that at the output a 1 is observed?

5.b)

What is the probability that a 1 was the output of the source if at the output of the channel a 1 is observed?

Problem 6

Suppose X

and Y

are each Gaussian random variables with means μ X

μ X

and μ Y

μ Y

and variances σ X 2

σ X 2

and σ Y 2

σ Y 2

. Assume that they are also independent. Show that Z=X+Y

Z X Y

is also Gaussian. Find the mean and variance of Z

Comments, questions, feedback, criticisms?

Send feedback

More about this content: Metadata | Version History | Cite This Content

This work is licensed by Behnaam Aazhang. See the Creative Commons License about permission to reuse this material.

Last edited by Charlet Reedstrom on Oct 25, 2004 10:41 am GMT-5.

Connexions