Properties of Mathematical expectation
(i) E(a) = a , where ‘a’ is a constant
(ii) E(aX) = aE(X)
(iii) E (aX + b) = aE(X ) + b , where ‘a’ and ‘b’ are constants.
(iv) If X ≥ 0,then E(X)≥ 0
(v) V (a) = 0
(vi) If X is random variable, then V (aX + b) = a 2V(X)
The moments (or raw moments) of a random variable or of a distribution are the expectations of the powers of the random variable which has the given distribution.
If X is a random variable, then the rth moment of X , usually denoted by μr , is defined as
provided the expectation exists.
If X is a random variable, the rth central moment of X about a is defined as E[(X − a)r ]. If a = μx , we have the rth central moment of X about μx , denoted by μr , which is
μr = E[(X–μX)r]
· μ'1 = E(X) = μX , the mean of X.
· μ1 = E[X– μX] = 0.
· μ2 = E[(X– μX)2], the variance of X .
· All odd moments of X about μX are 0 if the density function of X is symmetrical about μX , provided such moments exist.