Answer to Question #286815 in Statistics and Probability for jay

Question

In a communication system each data packet consists of 2000 bits. Due to the noise, each bit may be received in error with probability 0.1. It is assumed bit errors occur independently. Find the probability that there are more than 240 errors in a certain data packet.

Answer 1

If $X$ is a binomial random variable with mean $μ = np$ and variance $σ^2 = npq,$ then the limiting form of the distribution of

$$Z=\dfrac{X-np}{\sqrt{npq}},$$

as $n\to\infin,$ is the standard normal distribution $n(z; 0, 1).$

In practice, the approximation is adequate provided that both $np\geq 10$ and $nq\geq 10,$ since there is then enough symmetry in the underlying binomial

distribution.

Given

$n=2000, p=0.1$

$np=2000(0.1)=200>5,$

$nq=2000(1-0.1)=1800>5$

$X\sim N(\mu, \sigma^2)$

$$\mu=np=2000(0.1)=200$$

$$σ^2 = npq=2000(0.1)(1-0.1)=180$$

$$P(X>240)\approx1-P(X\leq 240)$$

$$=1-P(Z\leq\dfrac{240-200}{\sqrt{180}})\approx1-P(Z\leq2.981424)$$

$$\approx0.00143456$$