# Normal approximation for the Binomial distribution

The probability distribution function for Binomial distribution is

Assume

Stirling’s approximation for n!:

Use eq (2) with eq (1)

Equation (7) has two terms.

Term 1:

Term 2

Let’s work with Term 1 and reduce it further.

We need simplifications and series expansions.

Assume . Remember is the expected value of the Binomial distribution. So,  is simply looking at x as some deviation c from the mean or expected value of the distribution.

Now the approximate expansion (up to the second term) of

Using this, we can represent  as

In a similar fashion, we can reduce the second log term as follows.

Using expansion, we can write

We can substitute these two approximations in Term 1.

, assuming the second term will vanish as

So,

and,

Because, ,

Now, let’s work with Term 2.

Term 2

As , we can assume that the second term in the denominator vanishes leaving,

Substituting these two terms (approximation for Term 1 and approximation for Term 2) in equation (7), we get

For the binomial distribution, the expected value and the variance .

Using these notations with equation (8), we get the an approximation for the Binomial distribution.

or,

Equation (10) is the probability density function of the normal distribution (the bell shape).