Let for r = 1, 2,…,

????(r) = E[X(X − 1) · · · (X − r + 1)]

be the factorial moments of the random variable X, and

????r = [E(X − ????)r], ????′

r = E(Xr), r = 1, 2,…, be the moments of X around its mean, ???? = E(X) (central moments), and around zero, respectively.

Using properties of expectations, show that the following are true:

(i) ????′

1 = ????(1), ????1 = 0;

(ii) ????′

2 = ????(2) + ????(1), ????2 = ????′

2 − (????′

1 )2;

(iii) ????′

3 = ????(3) + 3????(2) + ????(1), ????3 = ????′

3 − 3????′

2????′

1 + 2(????′

1 )3;

(iv) ????′

4 = ????(4) + 6????(3) + 7????(2) + ????(1), ????4 = ????′

4 − 4????′

3????′

1 + 6????′

2(????′

1 )2 − 3(????′

1)4.

Verify also the truth of the identity Var(X) = ????(2) + ???? − ????2.