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* ^{2}*V*(*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 *r ^{th}* moment of

provided the
expectation exists.

If *X* is a
random variable, the *r ^{th}* central moment of

μ_{r} = *E*[(*X*–μ* _{X}*)

**Note**

·
μ'_{1}** **=** ***E*(*X*) =** **μ_{X}** **, the mean of** ***X*.

·
μ_{1}** **=** ***E*[*X*– μ* _{X}*] = 0.

·
μ_{2}** **=** ***E*[(*X*– μ* _{X}*)

·
All odd moments of X about** **μ_{X}** **are 0 if the density function of** ***X*** **is** **symmetrical about μ* _{X}* , provided such
moments exist.

Tags : Definition, Formulas , 12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation

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

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