×
Log in to StudySoup
Get Full Access to Statistics - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Statistics - Textbook Survival Guide

A binary message m, where m is equal either to 0 or 1, is

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 23E Chapter 4.5

Statistics for Engineers and Scientists | 4th Edition

  • Textbook Solutions
  • 2901 Step-by-step solutions solved by professors and subject experts
  • Get 24/7 help from StudySoup virtual teaching assistants
Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

4 5 0 258 Reviews
17
4
Problem 23E

A binary message m, where m is equal either to 0 or 1, is sent over an information channel. Because of  noise in the channel, the message received is X, where X = m + E, and E is a random variable representing the channel noise. Assume that if X ≤ 0.5 then the receiver, concludes that m = 0 and that if X > 0.5 then the receiver concludes that m = 1. Assume that E ~ N (0,0.25).

a. If the true message is m = 0, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 1?

b. Let σ2 denote the variance of E. What must be the value of σ2 so that the probability of error when m =0 is 0.01?

Step-by-Step Solution:
Step 1 of 3

Solution 23E

Step1 of 3:

We have a random variable X it presents the number of messages received.

Let X = m + E

Where,

E = a random variable representing the channel noise.

EN(0, 0.25)

m = Binary message.

If X 0.5 then the receiver concludes that m = 0

If X > 0.5 then the receiver concludes that m = 1

Here our goal is:

a).We need to find the probability of an error when true message is m = 0 and also find the probability that the receiver concludes that m = 1.

b).We need to find the value of so that the probability of error when m = 0 is 0.01 where   denote the variance of E.

Step2 of 3:

 

a).

If m = 0 then EN(0, 0.25).

The probability of an error is given by

   P(Error) = P(X > 0.5)

Let,

P(X > 0.5) =1 -  P(Z < )

                  =1 -  P(Z < )

                  = 1 - P(Z < 1)

Here P(Z < 1) is obtained from standard normal table(area under normal curve)

= 0.8413 (In statistical table we have to see row 1.0 under column 0.00)

 Now,

P(X > 0.5) = 1 - P(Z < 1)

                  = 1 - 0.8413

                  = 0.1587

Hence, P(Error) = 0.1587.

Step3 of 3:

 

b).

If m = 0 then X= EN(0,...

Step 2 of 3

Chapter 4.5, Problem 23E is Solved
Step 3 of 3

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

Unlock Textbook Solution

Enter your email below to unlock your verified solution to:

A binary message m, where m is equal either to 0 or 1, is

×
Log in to StudySoup
Get Full Access to Statistics - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Statistics - Textbook Survival Guide
×
Reset your password