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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

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

Statistics for Engineers and Scientists | 4th Edition

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Problem 23E

A binary message \(m\), where \(m\) is equal either to 0 or to 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 \(X \leq 0.5\) then the receiver concludes that \(m=0\) and that if \(X>0.5\) then the receiver concludes that \(m=1\). Assume that \(E \sim N(0,0.25)\)

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 \(\sigma^{2}\) denote the variance of \(E\). What must be the value of \(\sigma^{2}\) so that the probability of error when \(m=0\) is 0.01 ?

Step-by-Step Solution:

Step 1 of 4

 

We have a random variable \(X\) that presents the number of messages received.

 

                                                         \(\text { Let } X=m+E\)

 

Where,

\(E=\) a random variable representing the channel noise.

\(E \sim N(0,0.25)\)

\(\mathrm{m}=\) Binary message.

If \(X \leq 0.5\) 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 the 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 \(\sigma^{2}\) so that the probability of error when \(\mathrm{m}=0\) is \(0.01\) where \(\sigma^{2}\) denote the variance of \(\mathrm{E}\).

 

Step 2 of 4

Chapter 4.5, Problem 23E is Solved
Step 3 of 4

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

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