Variance

The Weighted Means described by the distribution of a Random Variable. It is the 2nd degree Central Moment. Describes spead/average of squared distances from mean.

Formula

$V(X)=E((X-E(X){)}^2)=E(X^2)-E(X{)}^2$

• $\mu$ is $E(X)$ Mean

Alternate Formula

Square of the Standard Deviation. Aka SD without the square root. $V(X)={\sigma }^2$

Specifics

• Law of Total Variance

Properties

• $V(aX+bY+c)=a^{2}V(X)+b^{2}V(Y)$
• Variance of linear combinations: $V(X+Y)=V(X)+V(Y)+2Cov(X,Y)$
• $V(c)=0$ where $c$ is a constant

Example

• Sample of two people
• Probability $60\%$ that a person approves of the president
• Random variable describing how many people approve
• Find expectation. First, Random Variable $X\mapsto \{0,1,2\}$ We can write out the distribution:

x	p(x)
0	$0.4\ast 0.4=0.16$
1	$0.4\ast 0.4\ast 2=0.48$
2	$0.6\ast 0.6=0.36$
We find Expectation $E(X)=0.16\ast 0+0.48\ast 1+0.36\ast 2=1.2$
$E[(X-\mu ){]}^2={\sum }_{i=1}^n(x_i-\mu {)}^{2}p(x_i)$
$=(-1.2{)}^2(0.16)+(-0.2{)}^2(0.48)+(0.8{)}^2(0.36)=0.48$