Rate of packet arrival

Question

Channel has bit error probability p. Number of bits in packet is Poisson distributed with mean lambda. How do I calculate the rate of successful packet arrival? This is what I have done so far: Assume n bits in packets. k out of n bits are in error and arrive with prob p and rest n-k bits arrive with prob 1-p. How does the Poisson distribution come into the picture?

Answer 1

The number of bits in packet is poisson distributed with mean $\lambda$.

The number of bit errors ( X) in the packet = $\lambda p$

The rate of successful packet arrival will be P(X=0) = $\frac{(\lambda p)^0 \exp{(-\lambda p)}}{0!} = \exp(-\lambda p)$

Note:There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and np ≤ 10.