e-Statistics

Normal Approximation

Suppose that X is a binomial random variable with n = and p = . If the size n is adequately large, then the distribution of X can be approximated by the normal distribution with mean np = and standard deviation $\sqrt{np(1-p)}$ = . That is, a normal distribution approximates a binomial distribution. A general rule for “adequately large” n is to satisfy $np \ge 5$ and $n(1-p) \ge 5$ .

Let Y be a normal random variable whose distribution approximates the binomial distribution of a random variable X. Then the probability involving X can be approximated by that of Y. Having taken into account the fact that Y is a continuous random variable, the approximated probability

P(i = $\le X \le j =$ ) $\approx P(i-0.5 \le Y \le j+0.5)$ = .

is calculated with continuity correction.