若 A1,A2,A3 相互独立,则 A1 与 A2∪A3 相互独立。
证明
P(A1(A2∪A3))=P(A1A2∪A1A3)
=P(A1A2)+P(A1A3)−P(A1A2A3)
=P(A1)P(A2)+P(A1)P(A3)−P(A1)(A2A3)
=P(A1)[P(A2)+P(A3)−P(A2A3)]
=P(A1)P(A2∪A3)
Discrete Distributions
Binomial Distribution
Bin(k;n,θ)=(nk)θk(1−θ)n−k,k=0,...,n
Bernoulli Distribution
Ber(x;θ)=θk(1−θ)1−k,k=0,1
Multinomial Distribution
Mu(x|n,θ)=(nx1,⋯,xK)∏Kj=1θxjj,x=⎛⎝⎜⎜x1⋮xK⎞⎠⎟⎟
Multinoulli Distribution
Mu(x|1,θ)=∏Kj=1θxjj
Poisson Distribution
Poi(k|λ)=e−λλkk!
Empirical Distribution / Measure
Pemp(A)=1|D|∑x∈DΠ(x∈A)
Continuous Distributions
Gaussian Distribution
N(x;μ,σ2)=12πσ2−−−−√e−12σ2(x−μ)2
Φ(x;μ,σ2)=∫x−∞N(t;μ,σ2)dt
Φ(x;μ,σ2)=12[1+erf(z/2√)],where
z=x−μσ
erf(x)=2π√∫x0e−t2dt
Student t Distribution
T(t;μ,σ2,n)=1nπ−−−√Γ((n+1)/2)Γ(n/2)[1+1n(x−μσ)2]−n+12
F Distribution
U∼χ2(n1),V∼χ2(n2)
F=U/n1V/n2∼F(n1,n2):
f(x;n1,n2)=1B(x;n12,n22)(n