Consider again a communication system that transmits bits with independent proba

Question

Consider again a communication system that transmits bits with independent probability of error p = 0.001, with each group of eight bits corresponding to a seven-bit ASCII code with an 8^th bit checksum. Suppose that the number of undetected errors in a transmission of n eight-bit words is modeled as a Poisson random variable X, with parameter lambda) Compute E[X] for n = 1; 10; 100; 1000; 10000. b) Compute the probability that a transmission will contain at least one undetected error using both the binomial and the Poisson pmf for n = 1; 10; 100; 1000; 10000.

Explanation / Answer

a) For n=1 , E(X)=np=1*0.001=0.001

For n=10 , E(X)=np=10*0.001=0.01

For n=100 , E(X)=np=100*0.001=0.1

For n=1000 , E(X)=np=1000*0.001=1

For n=10000, E(X)=np=10000*0.001=10

b) We have to find P(X>=1)=1-P(X=0). These are calculated using binomila pmf as follows:

n P(X=0) P(X>=1) Poisson pmf excel function 1 0.9990 0.0010 =POISSON(0,0.001,FALSE) 10 0.9900 0.0100 =POISSON(0,0.01,FALSE) 100 0.9048 0.0952 =POISSON(0,0.1,FALSE) 1000 0.3679 0.6321 =POISSON(0,1,FALSE) 10000 0.0000 1.0000 =POISSON(0,10,FALSE)

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