manage your expectation

Manage Your Expectations

작성자: 노재현발표일: 2012-05-19

12년 5월 19일 토요일

Summary

선형변형 확률모형의 경우 E(aX) 는 aE(X)와 같이 선형 변환되고, Var(aX) 는 a^2Var(X)와 같이 제곱으로 선형변경됨.

확률독립사건은 + - 의 경우는 총합(Sigma) 연산이므로 E(X + Y) = E(X) + E(Y) 로 분리할 수 있다.


FormulasStatistic Shortcut or formula

E(aX + b) aE(X) + b

Var(aX + b) a^2 Var(X)

E(X) Sigma x P(X=x)

E(f(X)) Sigma f(x) P(X=x)

Var(aX - bY) a^2 Var(X) + b^2 Var(Y)

Var(X) E(X-mu)^2 = E(X^2) - mu^2

E(aX - bY) aE(X) - bE(Y)

E(X1 + X2 + X3) 3 E(X)

Var(X1 + X2 + X3) 3 Var(X)

E(X^2) Sigma x^2 P(X=x)

Var(aX - b) a^2 Var(X)


Fat Dan’s Casino(-0.77)

E(X) = -0.77

Var(X) = 2.69

0

0.25

0.50

0.75

1.00

-1 4 9 14 19

사건x 에 대한 환률분포

P(X=x)

Combination None Lemon Cherries $s/Cherry $$$ SumxP(X=x)E(X)x P(X=x)Var(X)(x-E(X))^2 P(X=x)

-1 4 9 14 190.977 0.008 0.008 0.006 0.001 1-0.977 0.032 0.072 0.084 0.019 -0.77

0.0516833 0.1820232 0.7636232 1.3089174 0.3908529 2.6971


Fat Dan’s Casino(-0.85)

E(X) = -0.85

Var(X) = 67.4275

Combination None Lemon Cherries $s/Cherry $$$ SumxP(X=x)E(X)x P(X=x)

Var(X)(x-E(X))^2 P(X=x)

-2 23 48 73 980.977 0.008 0.008 0.006 0.001 1-1.954 0.184 0.384 0.438 0.098 -0.85

1.2920825 4.55058 19.09058 32.722935 9.7713225 67.4275

0

0.25

0.50

0.75

1.00

-2 23 48 73 98

사건x 에 대한 환률분포

P(X=x)


Linear Transform

Y = 5X + 3

E(Y) = 5 E(X) + 3

Var(Y) = 5^2 Var(X)

Var(aX +b) = a^2 Var(X)


Independent Observation

E(X1 + X2 + ... Xn) = nE(X)

Var(X1 + X2 + ... Xn) = nVar(X)

X1 + X2 is not the same as 2X.

compare Var(X1 + X2) vs Var(2X)


Transform or Observation?(Q)

Linear transform

Independent obseration

The amount of coffee in an extra large cup of coffee;X is the amount of coffee in a normal-size cup o

Drinking an extra cup of coffee per day;X is the amount of coffee in a cup o

Finding the net gain from buying 10 lottery tickets;X is the net gain of buying 1 lottery ticket o

Finding the net gain from a lottery ticket after the price of tickets goes up;X is the net gain of buying 1 lottery ticket. o

Buying an extra hen to lay eggs for breakfast;X is the number of eggs laid per week by a certain breed of hen o


New slot machineon the block(Y)

E(X - Y) = E(X) - E(Y)

Var(X - Y) = Var(X) + Var(Y)

E(aX + bY) = aE(X) + bE(Y)

Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)

E(aX - bY) = aE(X) - bE(Y)

Var(aX - bY) = a^2 Var(X) + b^2 Var(Y)


Exercise


manage your expectation

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