Unions, Intersection, Complement
Conditional Probability
correlation, the postive one and the negative one (mutually exclusive events).
Independent Events
Independent doesn’t mean there is no intersection. It means the frequency of A in B is equal in all sample space, also the complement part as well.
The above can also be used to check independent events.
De Morgan’s laws
Discrete Probablity Distribution
the Expected Value and Variance
Bernouli Distribution
P(success X=1)=p P(failure X=0)=1-p
E(X)=p Var(X)=E(X^2)-E(X)^2 = p-p^2=p(1-p)
Bonimial Distribution
n independent Bernoulli distribution
n=2 E(X)=2p(1-p)+2P^2=2p Var(X)=2p(1-p)+4P^2-(2p)^2=2p(1-p)
X~Bonimial(n,p) E(X)=np var(X)=np(1-p)
Hypergeometric Distribution
sampling is done without replacement,e.g. drawn 4 red balls from 5 randomly drawn with replacement (all contains 6 red balls and 14 yellow balls)
N=20 the number of balls
n=6 the number of drawn balls, of objects samples
a=5 of successes in the population
The binomial distribution can sometimes provide a reasonable approximation to the hypergeometrix distribution (sampling less than around 5% of the population). The reason is, drawning a small number from big population alternate a little frequency.
Geometric Distribution
X is the number of getting the first success.
