Quod Erat Demonstrandum

2010/03/27

normal approximation to binomial probabilities

那天上八堂(兩堂中四,兩堂中六,四堂對卷),加午飯及放學開會,再加之前凌晨時份「衝改」模擬試卷(遲下 OSM,即 onscreen marking,老師限時限刻限地點改卷,不能再在凌晨[對於我來說是很好的工作時間]工作了),再再加上有兄弟學校的學生在當天做交換生上我的應數堂,理應準備一些好東西;但一切都因在下極度疲憊忙碌,完全沒有備課;導致當天我想用電腦顯示以 normal distribution 近似 binomial distribution 時,才知學校電腦不能安裝 Java 而不能看到這個模擬。可惜。不過當天一如既往,有笑聲地「愉快學習」,他們沒有察覺我有什麼異樣(我估)。

談到 normal approximation to binomial probabilities,我亂舉例:

擲公平銀仔(fair coin) 1000 次,得公(Head)的次數為 500 次以上的機會是什麼?當然,我立即說這是無聊的,有誰真的去擲 1000 次?最多以電腦模擬云云。

有學生答:500 \times (\frac{1}{2})^{1000}

雖然不對,但我很感謝他,因為要高年級的同學答(非無聊的)問題,是很難的。

我寫出了答案

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

隨即道,這樣做很麻煩,希望帶出 normal approximation 的好處。但心中立即想,這也不是難,由二項係數(binomial coefficients)的對稱性,不難得出

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

嘗試用 normal approximation,即 N(1000 \times 0.5, 1000 \times 0.5 \times 0.5),要求的概率為

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

嗯,不過是準確到小數點後第 3 個位而已。

那麼 normal approximation to binomial probabilities 的近似可以有多「好」?這我仍未(不知如何)研究。

由上面的討論可以無聊地用來出題,比如:

證明當正整數 n 值足夠大,我們有以下近似

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 | 回覆

    • 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 | 回覆

      • Thank you Fred for your comments.

        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


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