推 zeromagi:高手 04/27 11:29
※ 引述《MyLPOGC (OGCing)》之銘言：
: Let X and Y have the pdf f(x,y) = 1 , 0 < x < 1 , 0 < y < 1 ,zero elsewhere.
: Find the cdf and pdf of the product Z = XY ?
by convolution
1 1
fz(z)=∫fx,y(z/y,y)(1/y)dy=∫(1/y)dy=-lnz , 0<z<1
z z
( 0<z/y<1 and 0<y<1 =>z<y<1 )
z
Fz(z)=∫-lnz dz=z-zlnz
0
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題外話 convolution 最基本分成四種
A=X+Y
f(a)=∫f(a-y,y)dy
V=X-Y
f(v)=∫f(v+y,y)dy
W=XY
f(w)=∫f(w/y,y)*1/|y|dy
Z=X/Y
f(z)=∫f(yz,y)*|y|dy
大致分成這四種 (積分範圍要注意就是了)
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※ 編輯: mangogogo 來自: 218.175.188.207 (04/27 11:26)