Thursday, November 10, 2016

statistics - Change of Variables in Second Moment


X is a non-negative continuous random variable with pdf f(x). G(t)=tf(x)dx. Show that E[X2]=20tG(t)dt

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I tried to write out E[X^2] (second moment of X) and see if that corresponds to the G(t) equation. I wasn't able to do that well because I wasn't sure of how the change of variables works.


Answer



I'm assuming E[X2]<. Integration by parts gives


2b0tG(t)dt=b2G(b)b0t2G(t)dt=b2G(b)+b0t2f(t)dt.


Taking the limit as b results in


20tG(t)dt=0t2f(t)dt=E[X2].


To see that limbb2G(b)=0, note that for b>0,


E[X2]=b0x2f(x)dx+bx2f(x)dxb0x2f(x)dx+b2G(b).



So


0b2G(b)E[X2]b0x2f(x)dx.


The right-hand side tends to 0 as b, so limbb2G(b)=0.


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