A binary message m , where m is equal either to 0 or to 1, is sent over an infor

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 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, 0.24).

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 2 denote the variance of E. What must be the value of 2 so that the probability of error when m = 0 is 0.01?

Explanation / Answer

binary message m,

m = 0 or 1

X = m + E

E ~ N(0, 0.24)

(a) If true messsage is m = 0 then probability of an error that receiver concludes that m = 1.

so if E ~ N(0, 0.24)

when m = 0 , X ~ N(0, 0.24)

Pr(m > 0.5; 0, 0.24) = 1 - Pr(m =< 0.5; 0 ; 0.24)

Z = (0.5 - 0)/ sqrt(0.24) = 0.5/0.49 = 1.02

so Pr(m > 0.5; 0, 0.24) =1 - 0.8461 = 0.1539

so probability of an error that the receiver conclues that m =1 is 0.1539

(b) Now, the error probability is given which is 0.01

Pr(m > 0.5 ; 0; 2 ) = 0.01

Z - value = 2.33

0.5/ = 2.33

= 0.5/2.33 = 0.2146

2 = 0.046

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