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Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 4.5 - Problem 23e
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 4.5 - Problem 23e

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

ISBN: 9780073401331 38

Solution for problem 23E Chapter 4.5

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

Step 3 of 4

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