# Connexions

You are here: Home » Content » Digital Communication Systems » Homework 1 of Elec 430

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

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

##### What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

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.
• Rice Digital Scholarship

This collection is included in aLens by: Digital Scholarship at Rice University

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Behnaam Aazhang. E-mail the author

# Homework 1 of Elec 430

Module by: Behnaam Aazhang. E-mail the author

Summary: First homework of Elec 430

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

## Exercise 1

The current II in a semiconductor diode is related to the voltage VV by the relation I=eV1 I V 1 . If VV is a random variable with density function f V x=12e|x| f V x 1 2 x for <x< x , find f I y f I y ; the density function of II.

## Exercise 2

### 2.a)

Show that if AB={} A B {} then PrAPr B c A B c

### 2.b)

Show that for any AA, BB, CC we have PrABC=PrA+PrB+PrCPrABPrACPrBC+PrABC A B C A B C A B A C B C A B C

### 2.c)

Show that if AA and BB are independent the PrA B c =PrAPr B c A B c A B c which means AA and Bc Bc are also independent.

## Exercise 3

Suppose XX is a discrete random variable taking values 012n 0 1 2 n with the following probability mass function p X k={n!k!(nk)!θk1θnk  if  k=012n0  otherwise   p X k n k n k θ k 1 θ n k k 0 1 2 n 0 with parameter θ 0 1 θ 0 1

### 3.a)

Find the characteristic function of XX.

### 3.b)

Find X- X and σ X 2 σ X 2

### Hint:

See problems 3.14 and 3.15 in Proakis and Salehi

## Exercise 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 ω an even number appears A ω an even number appears ω and B= ω 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!)

## Exercise 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?

## Exercise 6

Suppose XX and YY 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 ZZ.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

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

##### What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

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

##### What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks