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# nonnegative random variable

請問當X is a nonnegative random variable. pdf is symmetric about a point a.

期望值 E[X] = [1-F(x)] dx積分,範圍從0~無限大

可以告訴我過程怎麼算嗎?

### 1 Answer

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- Lotus_本願山彌陀講堂Lv 61 decade agoFavorite Answer
1. F(x)=∫[0~x] f(t)dt, F'(x)=f(x)

2. ∫[0~∞] [1-F(x)]dx (分部積分)

= [1-F(x)]x |[0~∞] - ∫[0~∞] x d[1-F(x)]

=0- ∫[0~∞] x(-f(x)) dx

=∫[0~∞] xf(x) dx

=E(x)

註: The condition, f(x) is symmetric about a point a ,

means that f(x)=0 for sufficient large x, so that lim(x->∞) [1-F(x)]x =0

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