# Quod Erat Demonstrandum

## 2010/03/27

### normal approximation to binomial probabilities

$\sum_{r = 501}^{1000}C_r^{1000}(\frac{1}{2})^{1000}$

$\sum_{r = 501}^{1000}C_r^{1000} = (\frac{1}{2})(2^{1000} - C_{500}^{1000}) \approx 0.487387491$ (資料由 EXCEL 提供)

$P(z > \frac{500.5 - 500}{\sqrt{250}}) \approx P(z > 0.031622777) \approx 0.48740000000000006$ (資料由以下網址提供)

http://www.psychstat.missouristate.edu/pdf/pdfj.htm

$C_n^{2n} + \sqrt{\frac{2}{\pi}}\int_{1/\sqrt{2n}}^\infty \exp(-\frac{x^2}{2})dx \approx 4^n$

P.S. 當天，還要加放學時學生問數，實在非常充實，感謝天父！

## 3 則迴響 »

1. 其實john sir你可以直接整histogram加normal density line, 再save幾張圖嚟示範, 無咁interactive但起碼唔會有tech problem(好似係)

習慣用normal, 應該係因為asymptotically binomial = normal
唔用john sir你講嗰個方法, 大槪係因為嗰個方法要"諗", 而且用嘅唔係statistician比較唔"熟悉"嘅嘢 (純屬吹水, 無比較過呢兩個approximation嘅efficiency)

岔開少少, thanks to CLT, 好多fundamental stat methods都based on normality assumption, 一大堆tests(e.g. z-test)同models(e.g. ANOVA)嘅inference都係, 所以話stat人最熟嘅distribution係normal都不為過

迴響 由 Fred — 2010/03/28 @ 2:31 上午 | 回應

• 另, 當p=/=0.5時, 用binomial coef做approximation係咪已經唔得呢?

迴響 由 Fred — 2010/03/28 @ 2:35 上午 | 回應

Yes, I could use steady pictures instead. But I just want to change the parameters of binomial distributions to see the corresponding changes in the shape of distributions so as to persuade students to believe the usefulness of normal approximation.

As you’d said that when $p \ne 0.5$, the method failed in general. Open question: can we find the exact probabilities by using binomial identities (say) for $p \ne 0.5$ ?

Students asked the validity of using normal approximation, i.e. what is the range of values of $n$ such that the use of normal approximation is GOOD enough? The remark in the textbook is $n > \max\{\frac{16p}{q} , \frac{16q}{p}\}$, but the reasoning seems to be not-so-strong. Or even, can we estimate the error bound of using normal approximation?

And of course, all these are non-sense in AL exam, it is just for further discussion only.

Thank you Fred again. Hope you and your friends will obtain brilliant academic results in University!

迴響 由 johnmayhk — 2010/03/28 @ 10:45 下午