I know that the normal can be used as an approximation to the binomial. I was looking for a derivation of this, and I found it via google in a math forum. Doctor Anthony begins:

What's with the k? Well, we're eventually going to want non-integer terms. The expansion of (q + p)

^{n}is familiar:

The i

^{th}term of the expansion is C(n,i).

OK. Notice use of the multiplication rule for variance from the other day.

Hmm... I know that

Lucie, you got some factoring to do. Let's deal with q and p first.

The left term has q

^{n-r-1}and the right term has q

^{n-r}, so we can factor out

q

^{n-r-1}, leaving a factor of q on the right-hand term in the brackets.

Similarly we can factor out p

^{r}from both sides leaving a factor of p on the left.

The combination expressions expand as shown above. We can factor out n! from both sides. We can factor out 1/(r+1)! from both sides, if we first multiply top and bottom of the right-hand term by (r+1), leaving (r+1) on the top.

Similarly, we can factor out (n-r)! from both sides, if we first multiply top and bottom of the left-hand term by (n-r), leaving (n-r) behind on the top. So everything checks out so far. Next, he wants to divide by y:

Hmm...again. We're dividing the expression we had above by y.

We have:

So both n! and (n-r)! terms cancel. We also cancel r!, leaving a factor of (r+1) on the bottom. The p

^{r}cancels, and the q

^{n-r}also cancels leaves a factor of q on the bottom. So I get:

Now we have to figure out how to rearrange the term in brackets:

Expand, and then substitute for p + q = 1:

It checks out. Doctor Anthony continues:

So far so good.

Go back to what

*we*had, and then multiply top and bottom by k

^{2}:

Hmm... The top is fine, but on the bottom we had

We need to get to:

He says:

OK, so we have:

Moving on to substitute for (np-r) k = -x on top and multiplying out on the bottom yields:

The only tricky part here was that we've replaced npqk

^{2}by s

^{2}.

Now he says:

And we're there! If we integrate the left side we get ln(y), and the right side is

-x

^{2}/ 2s

^{2}

y = A exp { -x

^{2}/ 2s

^{2}}

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