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# Properties of Mathematical expectation

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.

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)

## Concept of moments

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.

## Definition 6.11

If X is a random variable, then the rth moment of X , usually denoted by μr , is defined as provided the expectation exists.

### Definition 6.12

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[(XμX)r]

Note

·        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.

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12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation : Properties of Mathematical expectation | Definition, Formulas