Question

How do you find the normal approximation without using the continuity correction?

Answer 1

Let's say you want to compute the probability
`\mathbb P(a\le X\le b)` where
`X` converges in distribution to
`Y`, and
`Y` follows a normal distribution. The normal approximation (without the continuity correction) basically involves choosing
`Y` such that its mean and variance are the same as those for
`X`.

Example: If
`X` is binomially distributed with
`n=100` and
`p=0.1`, then
`X` has mean
`np=10` and variance
`np(1-p)=9`. So you can approximate a probability in terms of
`X` with a probability in terms of
`Y`:


`\mathbb P(a\le X\le b)\approx\mathbb P(a\le Y\le b)=\mathbb P\left(\frac{a-10}3\le\frac{Y-10}3\le\frac{b-10}3\right)=\mathbb P(a^*\le Z\le b^*)`

where
`Z` follows the standard normal distribution.