A binary symmetric channel (BSC) has an error probability Pe= 10° (ie., the prob

Question

A binary symmetric channel (BSC) has an error probability Pe= 10° (ie., the probability of receiving when 1 is transmitted, or vice versa, is Pe). Note that the channel behaviour is symmetrical with respect to 0 and 1. Thus, P(011) = P(1 10)-Pe P(O10) P(1|1)-1-P here Plylx)ldenotes the probability of receiving y when x is transmitted. A sequence of 8 binary digits is transmitted over this channel. (a). Determine the probability of receiving exactly 2 digits in error. (b). Repeat part (a) if a sequence of 6 binary digits is transmitted over the channel

Explanation / Answer

a)let X denotes the number of error digits received in transmission of 8 binary digits

and the probability of error is Pe

hence X~Bin(8,Pe)

so pmf of X is P[X=x]=8Cx(Pe)x(1-Pe)8-x   x=0,1,2,3,...,8

so P[receiving exactly 2 error digits]=P[X=2]=8C2(Pe)2(1-Pe)6=28*(Pe)2(1-Pe)6 where Pe=10-6

b) now if 6 binary digits are transmitted then it is

P[X=2]=6C2(Pe)2(1-Pe)6-2 [because now X~Bin(6,Pe)]

=15*(Pe)2(1-Pe)4   

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