Introduction to Probahili- the noise, each bit may be received in error with pro

Question

Introduction to Probahili- the noise, each bit may be received in error with probability 0.01. It is assumed that bit errors occur independen t. (30 points) In a communication system, each data packet consists of 1000 bits. Due to Define Xi as the indicator random variable for the ith bit in the packet, where X- if the ith bit is received in eror, and X-0 otherwise. Let Y be the total number of bit errors in the packet, ie, y. Xi+G + +Km. a) (1o pts) Find the expected value and variance of Y b) (i0 pty U'e them. more than 20 bit errors, Le, 20). limit theorem, approximate the probability that there are (10 pts) A data file need to be framed into multiple packets to be delivered. Suppose that the number of data packets needed to deliver a data file is a random variable N uniformly distributed between 1 and 10. c) Find the expected value and variance of the total number of bit errors in the packets that are used in delivering a data file.

Explanation / Answer

Y = X1 + X2 + ..X1000
Y follow binomial distribution with n = 1000 , p = 0.01

E(Y) = np = 1000*0.01 = 10
var(Y) = npq = 10 * 0.99 = 9.9

b)
P(Y > 20)
= P(Y > = 20.5)
=P(Z >(20.5 - 10)/sqrt(9.9))
= P(Z> 3.337119)
= 0.0004

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