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De Moivre Laplace theorem question

  • Thread starter stlukits
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  • #1
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Homework Statement



According to the de Moivre Laplace theorem

[tex]\binom{n}{k}p^{k}q^{n-k}\approx\frac{1}{\sqrt{2\pi{}npq}}e^{-\frac{(k-np)^{2}}{2npq}}[/tex]

For p=q=1/2 this translates nicely into an approximation for the binomial distribution by the normal distribution (the +1/2 is a continuity correction):

[tex]\binom{n}{k}\approx{}2^{n}N(\frac{n}{2},\frac{n}{4})(k)[/tex]

and therefore

[tex]\mbox{(A)}\quad \sum_{k=0}^{m}\binom{n}{k}\approx{}2^{n}\int_{-\infty}^{m+\frac{1}{2}}N(\frac{n}{2},\frac{n}{4})(x)dx[/tex]

This appears to be correct. I am trying to solve a problem for which I need

[tex]\sum_{k=0}^{m}k\binom{n}{k}[/tex]

and I am wondering what would keep me from reasoning in analogy to (A) so that

[tex]\mbox{(B)}\quad \sum_{k=0}^{m}k\binom{n}{k}\approx{}2^{n}\int_{-\infty}^{m+\frac{1}{2}}xN(\frac{n}{2},\frac{n}{4})(x)dx[/tex]

Unfortunately, when I use (B) I get some very counter-intuitive results (too complicated to expand on them here).

Homework Equations



see above

The Attempt at a Solution



see above
 
Last edited:

Answers and Replies

  • #2
22,097
3,278
Hi stlukits! :smile:

Let X have the binomial(n,1/2) distribution, and let Z have the normal(0,1) distribution. Let

[tex]Y_n=\frac{X-n/2}{\sqrt{n/4}}[/tex]

What you are using is that Yn converges in distribution to Z. This is correct and yields the good results you're getting.

However, after that you use something like

[tex]E[f(Y_n)]\rightarrow E[f(Z)][/tex]

For

[tex]f(x)=\left\{\begin{array}{c}x~\text{if}~x\leq m\\ 0~\text{if}~x>m\end{array}\right.[/tex]
However, this does not need to hold if you only know convergence in distribution. By Portmanteau theorem, this only holds for functions f with certain properties (like boundedness, which this function f is not).

However, not all is lost. You can do

[tex]\binom{n}{k}=\frac{n}{k}\binom{n-1}{k-1}[/tex]

and thus

[tex]\sum_{k=1}^m{k\binom{n}{k}}=n\sum_{k=1}^m{\binom{n-1}{k-1}}[/tex]

And now you can apply formula A again (which DOES work well!)
 
  • #3
56
3
Simple and brilliant. I'll try it. And thanks for explaining why (A) is legit.
 

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