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Chapter 5: Q.5.2 (page 215)

Show thatE[Y]=∫0∞PY>ydy-∫0∞PY<-ydy.

Hint: Show that ∫0∞PY<-ydy=-∫-∞0xfy(x)dx

∫0∞PY>ydy=∫0∞xfy(x)dx

Short Answer

Expert verified

Therefore,

E[Y]=∫0∞PY>ydy-∫0∞PY<-ydy

Hence Proved.

Step by step solution

01

Given Information.

E[Y]=∫0∞PY>ydy-∫PY<-ydy

02

Explanation.

E[Y]=∫-∞∞x.fy(x)dx=∫-∞0x.fy(x)dx+∫0∞x.fy(x)dx

=I1+I2

I1=∫-∞0x.fy(x)dx=-∫0∞x.fy(x)dx

=-∫0∞∫-x0dy.fy(x)dx

03

Explanation.

I1,-∞<-x<-y<0

I1=-∫0∞∫-∞yfy(x)dxdy

=-∫0∞P(Y<-y)dy

I2=∫0∞x.fy(x)dx

=∫0∞∫0∞dy.fy(x)dx,0≤y≤α≤∞

04

Explanation.

I2=∫0∞∫Y∞fy(x)dxdy

=∫0∞P(Y>y)dy

E(Y)=∫0∞PY>ydy-∫0∞PY<-ydy

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