in the sense that the ratio of the left-hand side to the right-hand side converges to 1 as n → ∞.
Proof
The theorem can be more rigorously stated as follows: , with a binomially distributed random variable, approaches the standard normal as , with the ratio of the probability mass of to the limiting normal density being 1. This can be shown for an arbitrary nonzero and finite point . On the unscaled curve for , this would be a point given by
For example, with at 3, stays 3 standard deviations from the mean in the unscaled curve.
The normal distribution with mean and standard deviation is defined by the differential equation (DE)
- with an initial condition set by the probability axiom .
The binomial distribution limit approaches the normal if the binomial satisfies this DE. As the binomial is discrete the equation starts as a difference equation whose limit morphs to a DE. Difference equations use the discrete derivative, , the change for step size 1. As , the discrete derivative becomes the continuous derivative. Hence the proof need show only that, for the unscaled binomial distribution,
- as .
The required result can be shown directly:
The last holds because the term dominates both the denominator and the numerator as .
As takes just integral values, the constant is subject to a rounding error. However, the maximum of this error, , is a vanishing value.[4]
Alternative proof
The proof consists of transforming the left-hand side (in the statement of the theorem) to the right-hand side by three approximations.
First, according to Stirling's formula, the factorial of a large number n can be replaced with the approximation
Thus
Next, the approximation is used to match the root above to the desired root on the right-hand side.
Finally, the expression is rewritten as an exponential and the Taylor Series approximation for ln(1+x) is used:
Then
Each "" in the above argument is a statement that two quantities are asymptotically equivalent as n increases, in the same sense as in the original statement of the theorem—i.e., that the ratio of each pair of quantities approaches 1 as n → ∞.