Basic Data Science : Statistical Reviews

Mean

$\mu =\sum _{i=1}^N(p_{i}N)=E(X)$

Rule of mean

R 1 : เมื่อ  C = constant,  $E(C)=C$

R 2 : เมื่อ  C = constant,  $E(X+C)=E(X)+C$

R 3 : เมื่อ  C = constant,  $E(CX)=C\times E(X)$

R4 :  $E(X+Y)=E(X)+E(Y)$


Variance

${\sigma }_x^2=\sum _{i=1}^N{p}_i[X_i-E(X){]}^2=VAR(X)$

หรือ

${\sigma }_x^2=E[(X_i-E(X){)}^2]$

หรือ

${\sigma }_x^2=E(X^2)-(E(X){)}^2$


Rule:

R1 : $Var(C)=0$

R2 : $Var(X+c)=E[X_i+c-E(X+c){]}^2=Var(X)$

R3 : $Var(cX)=c^{2}Var(X)$

R4: $Var(X+Y)=Var(X)+2COV(X,Y)+Var(Y)$

R5: ${\sigma }^2=\frac{\sum (x-\overline{x}{)}^2}{n}$

$n{\sigma }^2=\sum (x^2-2x\overline{x}+{\overline{x}}^2)$

$n{\sigma }^2=\sum x^2-2\overline{x}\sum x+\sum {\overline{x}}^2$

$n{\sigma }^2=\sum x^2-2n{\overline{x}}^2+n{\overline{x}}^2,\sum x=n\overline{x}$

$n{\sigma }^2=\sum x^2-n{\overline{x}}^2$

$\sum x^2=n{\sigma }^2+n{\overline{x}}^2$

$\sum x^2=n({\sigma }^2+{\overline{x}}^2)⇢\ast \ast \ast$