Question

What is convergence of binomial to and to notmal distribution?

Answer 1

Let $X$ be a Binomial random variable given as,

$f(x)=\binom{n}{x}p^xq^{n-x},\space x=0,1,2,.....n$

$0,\space elsewhere$

For small values of $n$ the probability that $X$assumes a specified value can be obtained from the Binomial tables. When $n$ is sufficiently large, and the probability of success $(p)$ is close to 0.5, the distribution of $X$ tends to the normal distribution with mean and standard deviation given by,

$\mu=E(x)=np$ and $\sigma=\sqrt{var (x)}=\sqrt{np(1-p)}=\sqrt{npq}$

We write,

$f(x)\sim N(np,(npq))$

To find $p(X=a)=p(a-0.5\lt X\lt a+0.5)$

$=p({a-0.5-np\over\sqrt{npq}}\lt Z\lt{a+0.5-np\over\sqrt{npq}})$ where 0.5 is a correction factor for continuity.

A possible guide to determine when the normal approximation may be used is provided by calculating $np$ and $n(1-p)$. If both of these quantities exceed 5, the approximation will yield good results.