The probability distribution function for Binomial distribution is
Stirling’s approximation for n!:
Use eq (2) with eq (1)
Equation (7) has two terms.
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
Now, let’s work with 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.
Equation (10) is the probability density function of the normal distribution (the bell shape).