作者mangogogo (mangogo)
看板Statistics
標題Re: [問題] 想請問幾個問題
時間Wed Nov 29 09:10:54 2006
※ 引述《yenway928609 (選擇~)》之銘言：
: 1.Let x and y be two r.v with uniform distribution and the interval [1,0]
: let z = x + y,Find the probability distribution of z
by convolution y=z-1
z | ╱︳
|╱==︳
fz(z)=∫f(y,z-y)dy (0≦y≦1 and z-1≦y≦z) |==╱︳y=z
|╱ ︳
(X Y independent ? ) ----------
z 1 y
case 1 : when 0≦z<1 fz(z)=∫ dy=z
0
1
case 2 : when 1<z≦2 fz(z)=∫ dy=2-z
z-1
thus fz(z)= z 0≦z≦1
2-z 1<z≦2
0 otherwise
: 2.prove gamma(1/2)= 根號(pi) (sorry不知道符號怎麼打~)
∞ -1/2 -t
gamma(1/2)=∫ t e dt
0
(let t=s^2 dt =2sds)
∞ -1 -s^2 ∞ -s^2
=∫ s e 2sds=2∫ e ds=2I
0 0
∞∞ -s^2-t^2
I^2=∫∫ e dsdt
0 0
then using polar transformation
we can get I=√π/2
: 3.let x be a r.v and y = g(x),prove
: a.e(y) = e[g(x)]
: b.f_y(y) = f_x(x1)/g'(x_1) + f_x(x2)/g'(x_2) +...f_x(x_n)/g'(x_n)
: where y = g(x_1) = g(x_2)=...
: 學長上午問我的.
: 謝謝.....
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◆ From: 140.113.114.192
※ 編輯: mangogogo 來自: 140.113.114.192 (11/29 09:16)