1. Manage Your
Expectations
작성자: 노재현
발표일: 2012-05-19

2. 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) 연
산이므로 E(X + Y) = E(X) + E(Y) 로 분리할 수
있다.

3. Formulas
Statistic 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)

4. Fat Dan’s Casino(-0.77)
Combination None Lemon Cherries $s/Cherry $$$ Sum
x -1 4 9 14 19
P(X=x) 0.977 0.008 0.008 0.006 0.001 1
E(X) -0.977 0.032 0.072 0.084 0.019 -0.77
x P(X=x)
Var(X) 0.0516833 0.1820232 0.7636232 1.3089174 0.3908529 2.6971
(x-E(X))^2 P(X=x)
사건x 에 대한 환률분포
1.00
0.75
E(X) = -0.77
0.50
Var(X) = 2.69
P(X=x)
0.25
0
-1 4 9 14 19

5. Fat Dan’s Casino(-0.85)
Combination None Lemon Cherries $s/Cherry $$$ Sum
x -2 23 48 73 98
P(X=x) 0.977 0.008 0.008 0.006 0.001 1
E(X) -1.954 0.184 0.384 0.438 0.098 -0.85
x P(X=x)
Var(X) 1.2920825 4.55058 19.09058 32.722935 9.7713225 67.4275
(x-E(X))^2 P(X=x)
사건x 에 대한 환률분포
1.00
E(X) = -0.85
0.75
0.50
Var(X) = 67.4275
P(X=x)
0.25
0
-2 23 48 73 98

6. Linear Transform
Y = 5X + 3
E(Y) = 5 E(X) + 3
Var(Y) = 5^2 Var(X)
Var(aX +b) = a^2 Var(X)

7. 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)

8. Transform or
Observation?(Q)
Linear Independent
transform obseration
The amount of coffee in an extra large cup of coffee;
o
X is the amount of coffee in a normal-size cup
Drinking an extra cup of coffee per day;
o
X is the amount of coffee in a cup
Finding the net gain from buying 10 lottery tickets;
o
X is the net gain of buying 1 lottery ticket
Finding the net gain from a lottery ticket after the price of tickets goes up;
o
X is the net gain of buying 1 lottery ticket.
Buying an extra hen to lay eggs for breakfast;
o
X is the number of eggs laid per week by a certain breed of hen

9. New slot machine
on 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)

10. Exercise