Normal distribution

設h = k/σ，則此方程會變成P(-2h < Z < h) = 0.9（亦由此證出P(-2k/σ < Z < k/σ) = 0.9的解是形如k = hσ，其中h是常數的形式）。

2010-03-05 16:09:43 補充：
用normal distribution table的解法：
找h1和h2使P(-2h1 < Z < h1) < 0.9 < P(-2h2 < Z < h2)，然後進行interpolation。但這解法的準確度較差。

2010-03-05 16:09:57 補充：
所以，較好的解法當然是用Newton's method（fix-point iteration似乎不太可行）解∫(-2h to h) 1/√2π e^(-x²/2) dx = 0.9，可用fx-3650P幫手，這至少可以解出一個準確至5位小數的近似解。（∫(-2h to h) 1/√2π e^(-x²/2) dx = 0.9亦是方程一條！）

唯一留意的是，d/dh(∫(-2h to h) 1/√2π e^(-x²/2) dx - 0.9)需打醒十二分精神，千萬不要d錯！

2010-03-09 07:01:32 補充：
nelsonywm2000，既然你能提出這個非常特別的方法，點解你仲唔直接回答這題？難道要STEVIE-G™揀垃圾回答嗎？

2010-03-11 08:36:38 補充：

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參考資料：
my wisdom of maths

thanks nelson...
等我詳細參閱一下~

可先以P(X<μ+k)=0.9得出k約數,再計算P(μ-2k<μ+k)的準確值,
再而調整k的估值,以iteraction方法得最近值.
若以Normal Distribution Table可得k大約於1.307及1.308之間.
匿名提供之Excel Goal Seek 也只是約數1.3074.其實可以Excel函數Norminv及Normdist
利用上面方法求得非常準確的估值.

2010-03-06 14:51:41 補充：
http://img408.imageshack.us/img408/9181/21536635.png

2010-03-13 13:53:23 補充：
Excel Goal Seek is a good method to solve the problem since in all practical cases for statistics, we do not need to find a very accurate answer. So in this sense, Excel Goal Seek is not at all a junk solution. It is simple. I am here to share my another thought and a precise figure I can get.

The method by ilovemadonna2009 is right, but the answer is wrong

Actually,the required range is not symetrical. How could we estimate the answer by symetry

With the goal seek function in excel, the numerical answer is 1.3074

2010-03-10 08:38:30 補充：
nelsonywm2000, your method (i.e. Newton's Method) is also one of the numerical method. It seems it not much better than using Excel to solve it.

Do you have any analytical method to find the exact answer? e.g. using integration?

2010-03-12 08:35:54 補充：
doraemonpaul, it seems that the difference is only the degree of accuracy. All we used / proposed / suggested are numerical methods. There are a thousand of different numerical methods available, depending on the accuracy and efficiency.

2010-03-12 08:36:11 補充：
unless there is an analytical solution

2010-03-12 08:39:07 補充：
Actually, you could use Simpson's Rule to evaluate the definite intregal, rather than use Newton's Method. it is much more direct and appropriate.