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

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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.
User Anujkk
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