It is obvious that if X and Y are independent and uniformly distributed on [a,b] random variables, and E[X]=E[Y]=m≡a+b2, then the random variable Z=(X−Y)sign(X−m) has the positive expectation, that is, E[Z]>0.
Indeed,
E[Z]=1(b−a)2∫ba∫ba(x−y)sign(x−m)dxdy=
=1b−a∫baxsign(x−m)dx−1(b−a)2∫basign(x−m)dx∫baydy=
=1b−a(∫bmxdx−∫maxdx)−mb−a∫basign(x−m)dx=
=1b−a(∫bmxdx−∫maxdx)−mb−a(∫bmdx−∫madx)=b−a4>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=(X−Y)sign(X−m). Is it true that E[Z]>0?
My calculations:
E[Z]=∫∞−∞∫∞−∞(x−y)sign(x−m)f(x)f(y)dxdy=
=∫∞−∞∫∞−∞xsign(x−m)f(x)f(y)dxdy−∫∞−∞∫∞−∞ysign(x−m)f(x)f(y)dxdy=
=∫∞−∞xf(x)sign(x−m)dx−m∫∞−∞f(x)sign(x−m)dx=
=∫∞mxf(x)dx−∫m−∞xf(x)dx−m(∫∞mf(x)dx−∫m−∞f(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=(X−Y)sign(X−m)? Is it true that E[Z]>0 in this case too?
In this case we have
E[Z]=∫∞−∞∫∞−∞(x−y)sign(x−m)f(x,y)dxdy=
=∫∞m∫∞−∞(x−y)f(x,y)dxdy−∫m−∞∫∞−∞(x−y)f(x,y)dxdy.
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