91Ó°ÊÓ

The distribution of a random variable X is symmetric with respect to the point \(x = 0\)and \(E\left( {{X^4}} \right) < \infty \).

It is given that the distribution of X is symmetric with respect to the point \(x = 0\). Thus, the mean of X is 0.

Now, it is known that the \(M.S.E = E\left( {{{\left( {X - d} \right)}^2}} \right)\)is minimized when d takes the value of the mean of X.

That is, \(E\left( {{{\left( {X - d} \right)}^2}} \right)\)is minimized when\(d = 0\).

Now, let\(X - d = a\).

If \(E\left( a \right)\) is minimized at \(d = 0\), then \(E\left( {{a^2}} \right)\)should also be minimized at \(d = 0\).

Now,

\(\begin{aligned}{}E\left( {{a^2}} \right) &= E\left( {{{\left( {{{\left( {X - d} \right)}^2}} \right)}^2}} \right)\\ &= E\left( {{{\left( {X - d} \right)}^4}} \right).\end{aligned}\)

Also, \(E\left( {{{\left( {X - d} \right)}^4}} \right) < \infty \)because \(E\left( {{X^4}} \right) < \infty \).

Hence, \(E\left( {{{\left( {X - d} \right)}^4}} \right)\)is minimized when \(d = 0\).