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Let's say you want to compute the probability 
where 
converges in distribution to 
, and follows a normal distribution. The normal approximation (without the continuity correction) basically involves choosing such that its mean and variance are the same as those for .
Example: If is binomially distributed with 
and 
, then has mean 
and variance 
. So you can approximate a probability in terms of with a probability in terms of :
where
follows the standard normal distribution.
Example: If
where