Another approach helpful to find the summary measures for probability distribution is based on the ‘moments’.

**Moments**

Another
approach helpful to find the summary measures for probability distribution is
based on the ‘moments’. We will discuss two types of moments.

i.
Moments about the origin. (Origin may be zero or any other constant say *A*
). It is also called as raw moments.

ii.
Moments about the mean is called as central moments.

First order central moment:

put r = 1 in the definition,

This is called the mean of the random variable X. Hence the
first order raw moment is mean.

Second order raw moment

Put r = 2 then

μ'_{2} = E(X^{2})= Σx_{i}^{2}p_{i}

This is called second order raw moment about the origin.

First order central moment:

put r = 1 in the definition,

This
is called the 2^{nd} central moment about the mean and is known as the
variance of the random variable *X.*

**i.e., Variance = Var
(X) =μ _{2} = μ_{2}' - (μ_{1}')^{2}**

**Standard Deviation
(S.D) = σ = √variance**

i.
Var (*c*) = 0 i.e. Variance of a constant is zero

ii.
If *c* is constant then Var (cX)=c^{2} Var(X)

iii.
If *X* is a random variable and *c* is a constant then Var (*X*
± *c*) = Var (*X*)

*iv. a *and* b *are constants then Var (*aX *±* b*)
=* a*^{2} Var (*X*)

*v. a *and b are constants then Var (*a *±* bX*)
=* b*^{2} Var (*X*).

vi.
If *X* and *Y* are independent random variables
then Var (*X* + *Y*) = Var (*X*)
+ Var(*Y*)

A moment
generating function (m.g.f) of a random variable X about the origin is denoted
by M_{x}(t) and is given by

M_{x}(t)
= E (e^{tx}) , |t| <1

From the series on the right hand side, μ_{r}’
is the coefficient of r^{t}/r! in Mx (t) .

For
a random variable X to find the moment about origin we use moment generating
function.

Since
*M _{x}* (t) generates moments of the
distribution and hence it is known as moment generating function.

Using
the function, we can find mean and variance by using the first two raw moments.

μ_{2}= μ_{2}’-( μ_{1}’)^{2}

For
some distribution, the M.G.F does not exist. In such cases we can use the
characteristic function and it is more servicable function than the M.G.F.

Tags : Definition, Formula, Solved Example Problems , 11th Statistics : Chapter 9 : Random Variables and Mathematical Expectation

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