[機統] 機率密度函數消失
Consider the following two pdfs,
1
f (x)=--------exp(-((logx)^2)/2), x≧0, and
1 √(2π)x
f (x)=f (x)[1+sin(2πlogx)], x≧0
2 1
Show that
(1)If X ~f (x), then E(X ^r)=exp((r^2)/2), r=0,1,2...
1 1 1
(2)Suppose X ~f (x). Then E(X ^r)=E(X ^r) for r==0,1,2...
2 2 1 2
第1小題已求出，問題是第2小題
第2小題的部分計算過程如下
∞ ∞
E(X ^r)=∫ (x^r)f (x)dx=∫ (x^r)f (x)[1+sin(2πlogx)]dx
2 0 2 0 1
∞ ∞
=∫ (x^r)f (x)dx+∫ (x^r)sin(2πlogx)f (x)dx
0 1 0 1
∞
=E(X ^r)+∫ (x^r)sin(2πlogx)f (x)dx
1 0 1
where
∞ ∞ 1
∫ (x^r)sin(2πlogx)f (x)dx=∫ (x^r)sin(2πlogx)--------exp(-((logx)^2)/2)dx
0 1 0 √(2π)x
( Let y=logx → x=exp(y)→ dx=exp(y)dy )
∞ 1
=∫ exp(ry)sin(2πy)-------exp(-(y^2)/2)dx
-∞ √(2π)
接下來就卡住了...
所以想請板上的高手幫忙解答
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※ 編輯: raymond168 來自: 203.70.109.92 (01/06 23:29)
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05/02 13:54, , 1F
05/02 13:54, 1F
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