Friday, March 9, 2018

probability theory - If X, Y are i.i.d. with mean m then E((XY)mathrmsign(Xm))>0

It is obvious that if X and Y are independent and uniformly distributed on [a,b] random variables, and E[X]=E[Y]=ma+b2, then the random variable Z=(XY)sign(Xm) has the positive expectation, that is, E[Z]>0.




Indeed,
E[Z]=1(ba)2baba(xy)sign(xm)dxdy=
=1babaxsign(xm)dx1(ba)2basign(xm)dxbaydy=
=1ba(bmxdxmaxdx)mbabasign(xm)dx=
=1ba(bmxdxmaxdx)mba(bmdxmadx)=ba4>0.



Now, let X and Y be independent and identically distributed random variables with the probability density function f(x), E[X]=E[Y]=m, and let Z=(XY)sign(Xm). Is it true that E[Z]>0?



My calculations:
E[Z]=(xy)sign(xm)f(x)f(y)dxdy=
=xsign(xm)f(x)f(y)dxdyysign(xm)f(x)f(y)dxdy=
=xf(x)sign(xm)dxmf(x)sign(xm)dx=
=mxf(x)dxmxf(x)dxm(mf(x)dxmf(x)dx).




And what will be if X and Y are not independent but identically distributed random variables with the joint probability density function f(x,y), E[X]=E[Y]=m; Z=(XY)sign(Xm)? Is it true that E[Z]>0 in this case too?



In this case we have
E[Z]=(xy)sign(xm)f(x,y)dxdy=
=m(xy)f(x,y)dxdym(xy)f(x,y)dxdy.

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