Question

In a communication system, information bits are transmitted from source to destination. However, due to the ambient white Gaussian noise in the communication channel, an information bit may be received erroneously by the time it arrives at the destination. Assume that the information bits are transmitted independently, and let p denote the bit error probability with pϵ(0; 1). Suppose a total of N bits are transmitted from the source to the destination, and let W_N denote the total number of bits received erroneously.

(a) What is the exact probability mass function (PMF) of W_N Find the expectation and variance of W_N.

(b) Suppose p = 10^-3 and N = 10⁷. Use the Central Limit Theorem to estimate the probability P(W_N ≥10,150).

Answer 1

a) As the information bits are transmitted independently, and the bit error probability is constant equal to p

So W_N will follow Binomial distribution with number of trials equal to N and probability of success ( in this case error) p.

Thus the PMF is

b) Using Central Limiting Theorem we know that W_N will follow normal distribution for large N, with mean equal to and standard deviation equal to

Hence, using normal table we have

using normal table