Mean
\(\large \mu = \sum_{i=1}^N(p_iN) = E(X) \)
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
\( \large \sigma^{2}_{x} = \sum_{i=1}^{N}p_i[X_i - E(X)]^2 = VAR(X) \)
หรือ
\( \large \sigma^{2}_{x} = E[(X_i - E(X))^2]\)
หรือ
\( \large \sigma^{2}_{x} = E(X^2) - (E(X))^2\)
Rule:
R1 : \( Var(C) = 0\)
R2 : \( Var(X+c) = E[X_{i}+c - E(X+c)]^2 = Var(X) \)
R3 : \( Var(cX) = c^2Var(X) \)
R4: \( Var(X+Y) = Var(X) + 2COV(X,Y) + Var(Y) \)
R5: \(\large \sigma^2 = \frac{\sum(x-\bar{x})^2}{n} \)
\(\large n\sigma^2 = \sum(x^2-2x\bar{x}+\bar{x}^2) \)
\(\large n\sigma^2 = \sum{x^2}-2\bar{x}\sum{x}+\sum{\bar{x}^2} \)
\(\large n\sigma^2 = \sum{x^2}-2n\bar{x}^2+n\bar{x}^2,\sum{x}=n\bar{x} \)
\(\large n\sigma^2 = \sum{x^2}-n\bar{x}^2 \)
\(\large \sum{x^2} = n\sigma^2 +n\bar{x}^2 \)
\(\large \sum{x^2} = n(\sigma^2 +\bar{x}^2)\dashrightarrow*** \)
\(\large \mu = \sum_{i=1}^N(p_iN) = E(X) \)
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
\( \large \sigma^{2}_{x} = \sum_{i=1}^{N}p_i[X_i - E(X)]^2 = VAR(X) \)
หรือ
\( \large \sigma^{2}_{x} = E[(X_i - E(X))^2]\)
หรือ
\( \large \sigma^{2}_{x} = E(X^2) - (E(X))^2\)
Rule:
R1 : \( Var(C) = 0\)
R2 : \( Var(X+c) = E[X_{i}+c - E(X+c)]^2 = Var(X) \)
R3 : \( Var(cX) = c^2Var(X) \)
R4: \( Var(X+Y) = Var(X) + 2COV(X,Y) + Var(Y) \)
R5: \(\large \sigma^2 = \frac{\sum(x-\bar{x})^2}{n} \)
\(\large n\sigma^2 = \sum(x^2-2x\bar{x}+\bar{x}^2) \)
\(\large n\sigma^2 = \sum{x^2}-2\bar{x}\sum{x}+\sum{\bar{x}^2} \)
\(\large n\sigma^2 = \sum{x^2}-2n\bar{x}^2+n\bar{x}^2,\sum{x}=n\bar{x} \)
\(\large n\sigma^2 = \sum{x^2}-n\bar{x}^2 \)
\(\large \sum{x^2} = n\sigma^2 +n\bar{x}^2 \)
\(\large \sum{x^2} = n(\sigma^2 +\bar{x}^2)\dashrightarrow*** \)
ความคิดเห็น
แสดงความคิดเห็น