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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=eV−1 I V 1 . If VV is a random variable with density function f V ⁢x=12⁢e−|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 A⁢B={} A B {} then PrA≤Pr B c A B c

2.b)

Show that for any AA, BB, CC 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 AA and BB are independent the PrA∩ B c =PrA⁢Pr 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 012…n 0 1 2 … n with the following probability mass function p X k={n!k!⁢(n−k)!⁢θk⁢1−θn−k  if  k=012…n0  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.

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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.

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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.

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