Moments and Other Measures in Terms of Expectations

Data Science and A.I. Lecture Series

By Bindeshwar Singh Kushwaha – PostNetwork Academy

Moments

The $r^{th}$ order moment about any point $A$ of a variable $X$ is given by:

For discrete variables:

$${\mu }_r^{\prime }=\sum _{i=1}^n{p}_i(x_i–A{)}^r$$

For continuous variables:

$${\mu }_r^{\prime }={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x–A{)}^{r}f(x)dx$$

Central Moments

The $r^{th}$ order central moment is given by:

For discrete variables:

$${\mu }_r=\sum _{i=1}^n{p}_i(x_i–\mu {)}^r$$

For continuous variables:

$${\mu }_r={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x–\mu {)}^{r}f(x)dx$$

Expectation form:

$${\mu }_r=E[(X–\mu {)}^r]$$

Variance

Variance of a random variable $X$ is the second-order central moment:

$$V(X)=E[X^2]–(E[X]{)}^2$$

Using moments about the origin:

$$V(X)={\mu }_2\prime –({\mu }_1\prime {)}^2$$

Theorem: Variance Scaling Property

If $X$ is a random variable and $a,b$ are constants, then:

$$V(aX+b)=a^{2}V(X)$$

Proof of Theorem

By definition of variance:

$$V(aX+b)=E[(aX+b–E[aX+b]{)}^2]$$

Expanding expectation:

$$=E[(aX+b–aE[X]–b{)}^2]$$

Simplifying:

$$=E[a^2(X–E[X]{)}^2]$$

Using expectation properties:

$$=a^{2}E[(X–E[X]{)}^2]=a^{2}V(X)$$

Example: Variance Calculation

X	p(X)
-2	0.15
-1	0.30
0	0
1	0.30
2	0.25

Computing $V(X)$:

$$V(X)=E[X^2]–(E[X]{)}^2$$

Using values:

$$V(X)=2.2–(0.2{)}^2=2.2–0.04=2.16$$

Variance of a Linear Transformation

Computing $V(2X+3)$:

Using theorem: $V(aX+b)=a^{2}V(X)$

$$V(2X+3)=4V(X)$$

Substituting $V(X)=2.16$:

$$V(2X+3)=4(2.16)=8.64$$

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