Definition (3 items) (can be used to prove whether it is an event domain)

Nature (3 items)

Event domain on real number space—Borel set class

Definition (3 items) (can be used to prove that it is a λ-type)

If the set class A is closed for the intersection operation, then λ(A) is closed for the intersection operation

multiplication formula

total probability formula

Bayesian formula

Multiple events are independent of each other

Multiple events are independent of each other

independent event family

Multiplication formulas and addition formulas for multiple events that are independent of each other

The necessary and sufficient condition for ξ to be a random variable mapping in Ω→R is ξ-1(B)∈F, for any B∈borel set

If ξ is a random variable mapped in Ω→R, then ξ-1(B) is a σ algebra, and ξ-1(B)⊂F

If the random variable sequence {ξn} converges to ξ, then ξ is a random variable

The composite mapping η=f(ξ)/can be extended to multivariate η=g(ξ1, ξ2,…,ξn)

multiple independent random variables

Independent random variable sequence/independent random variable family

The random variable family of ξ and the random variable η are independent of each other.

Integrate with joint distribution

The definition of simple random variables—ξ(Ω) is a finite set; standard expression

A nonnegative random variable ξ can be consistently approximated by a simple random variable ξn [Non-negative random variable approximation theorem]: If the random variable ξ≥0, Then there is a monoincreasing simple random variable sequence {ξn}, such that when n→∞, ξn=ξ

Any random variable can be approximated by simple random variables "The positive and negative parts of ξ are both non-negative random variables" [Random variable approximation theorem] Assuming that ξ is a real-valued function on Ω, then ξ is a necessary and sufficient condition for a random variable on (Ω, F, P) There is a simple random variable sequence {ξn} such that when n→∞, ξn=ξ

Definition/Density Matrix/Probability Support Set/Distribution and Distribution Function Formulas

Binomial distribution B(n,p)—k successes in n experiments b(k;n,p)—the most likely value of binomial distribution

Geometric distribution G(p)—number of first successful occurrences g(k;p)—no memory

Negative binomial distribution Nb(r,p)—the number of successful waits for the rth time f(k;r,p)

Poisson distribution P(λ)—ξt represents the number of particles p(k;λ) arriving in the (0, t] period Random choice invariance of the Poisson distribution/Most likely value of the Poisson distribution

Definition/Density Matrix/Probability Support Set/Distribution and Distribution Function Formulas

Uniform distribution U(a,b)—distribution function formula

Normal distribution N(a,σ²)—distribution function formula/properties of normal distribution/3σ principle

Γ-distributionΓ(λ,r)

Exponential distribution Γ(λ,1)—distribution function formula/no memory

The formula of joint distribution and joint distribution function/identity map/(Rn, Bn, Fξ) is a probability space

Joint distribution uniqueness theorem: distribution and distribution function are mutually uniquely determined

The properties of the joint distribution function F (4 items) include one more "non-negativity"

Discrete random vector

continuous random variable

Two-dimensional uniform distribution/two-dimensional normal distribution

Definition/Discrete—Edge Density/Continuity—Calculation of Edge Density Function/Edge Distribution Function

Definition of mutually independent random vectors

The necessary and sufficient condition for two random vectors to be independent of each other is that the joint distribution function variables are separable [If it is specific to discrete type or continuity, the joint density function variable can be separated]

Discrete full probability formula—when ξ and η are independent of each other, the discrete convolution formula can be used

Definition/mutually independent non-negative integer random variables ξ, η, Gξ η (s) = Gξ (s) Gη (s)

Distribution function/density function of a single continuous random variable function

Density formula of sum (continuous convolution formula)

Density formula of quotient

two random variables

Joint density function of continuous n-dimensional random variable ξ (replacement, J)

Chi-square distribution

Student's t distribution

F distribution

Use uniformly distributed random numbers to construct random numbers with discrete distribution density/construct exponentially distributed random numbers obeying λ=1/construct standard normal distributed random numbers

Definition (Simple Random Variable)—Properties can be proven based on the partitioning of simple random variables

Generalized definition (non-negative random variables)

Continue to promote (general random variables)