[數學] statistics(20點)

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.

Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles.

When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.

(let me write x1=x, x2=y for simplicity)

首先要明白問題問乜.

Actually this question has nothing to do with statistics, it is just an exercise on doing integration.



(a) 要證明佢係density, 就要證 Int Int F(x,y)dxdy = 1, 個integration 的range 係 0<x+y<1

(b) 要搵 X 的 Distribution, 即係問 X 唔 depend on Y 的時候的 distribution, 所以要"加"起所有Y. 所以題目係問: 證明 X~ Int F(x,y)dy = Beta(...) 同埋 Y~ Int F(x,y) dx = Beta (...)



好, 明了問題問乜, 就可以開始作答

但係我地仲要知個 range 0<x+y<1係乜.

畫幅圖, 就係 直角三角形 (0,0)~(1,0)~(0,1), (個角係(0,0))

所以 integrate 的時候, 係Int (0,1) Int(0,1-y) F(x,y)dxdy.



題目有比 hint, 跟住做:

(a) let u = x / (1-y). Then x = (1-y)u, dx = (1-y)du, when x=0, u=0; when x = 1-y, u = 1.

so we get (let C = that bunch of gamma function)



(I typed here:)

http://www.mathlinks.ro/Forum/latexrender/pictures/46bea6e12f8f884c7f2bb095a4068b3c.gif



(b) actually, what we want to find is just the same as the above integral, but only "single" integral (without dy)



from the 6-th line in part (a),

http://www.mathlinks.ro/Forum/latexrender/pictures/ec220f75cd27fa71ef4c8b886d21d13c.gif



do exactly the same thing for X.

(K is automatically normalized, because your F(x,y) is a density)

sample size n=20 is too small. It is recommended to use n>100.