Approximations using Poisson and Binomial distributions

Question

The probability of a bit error in a communication line is $10^{-5}$ per bit. Suppose we examine a string of $1000$ independent bits. Calculate the probability of $0$, $1$, $2$, and $3$ errors in the string using Poisson theory and using binomial theory. Compare the results.

Answer 1

Hints:

binomial distribution: $P(X=x)={n \choose x}\cdot p^x\cdot (1-p)^{n-x}$

$n=1000; p=10^{-5}, 1-p=1-10^{-5}$

poisson distribution: $P(X=x)=e^{-\lambda }\cdot \frac{\lambda ^x}{x!}$

$\lambda=n\cdot p=10^{-5}\cdot 1000=\frac{1}{100} $

Both distributions can be approximated by the normal distribution. In the case of a binomial distributed variable: $$P(X=x)=\Phi \left( \frac{x+0.5-n\cdot p}{\sqrt{n\cdot p \cdot (1-p)}} \right)-\Phi \left( \frac{x-0.5-n\cdot p}{\sqrt{n\cdot p \cdot (1-p)}} \right)$$

$\Phi(\cdot )$ is the cummulative function of the standard normal distribution

In the case of a poisson distributed variable: $$P(X=x)=\Phi \left( \frac{x+0.5-\lambda}{\sqrt{\lambda}} \right)-\Phi \left( \frac{x-0.5-\lambda}{{\sqrt{\lambda}}} \right)$$

There are some conditions for approximation. They are only rule of thumbs.

Binomial distribution: $n\cdot p\geq 10$ and $n\cdot (1-p)\geq 10$

Poisson Distribution: $\lambda >10$