Probability and random
variables
Events and Probabilities
Complement of event E
Intersection: both A and B occur
Union: in A or in B or in both
events
Mutually exclusive events互斥事件
Exhaustive events互补事件
Mutually exclusive and exhaustive events
e.g. the date you born
Assigning Probability
requirements
3 ways
0 ≤ P(Oi ) ≤ 1 for each i = 1,2,…,n
P(O1) + P(O2) + … + P(On) = 1
Classical approach
throw a die
Relative frequency approach
Subjective approach
depends on personal experience
Probabilities
margin
conditional
joint
P(A1 ∩ B1)
横列纵列相加，总和是1
Independence
P(A∩B) = P(A)P(B)
P(A|B) = P(A) and P(B|A) = P(B)
Rules of probability
Complement rule
Addition rule
If A and B are mutually exclusive
Multiplicative rule
Expected Values and Variance
EV(Mean)
Variance
2.4=mean
Laws
EV
Variance
Portfolio of Assets
covariance of two discrete random variables X and Y
Cov(X,Y) > 0
Positive linear relationship (i.e. X tends
to increase when Y increases)
Cov(X,Y) < 0
Negative linear relationship
Cov(X,Y) = 0
No linear relationship
coefficient of correlation
a fixed scale: -1 ≤ ρ ≤ 1
ρ = 1
perfect positive linear relationship
0 < ρ < 1
positive linear relationship (stronger as it approaches 1)
ρ = 0
no linear relationship
-1 < ρ < 0
negative linear relationship (stronger as it approaches -1)
ρ = -1
perfect negative linear relationship
sum