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A binary message m, where m is equal either to 0 or 1, is
Chapter 4, Problem 23E(choose chapter or problem)
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 ?
Questions & Answers
QUESTION:
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 ?
ANSWER: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 \(\mathrm{E}\).