Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas.

Understand combinatorics of probability distributions.

概率分布

Is the Poisson binomial distribution related to the Poisson distribution?
What is the event behind service time?
- Does it mean momentum, a particle appearing and then disappearing?
- Or the opposite, disappearing and then appearing?
- Or is it the same, just a matter of which direction?

https://en.wikipedia.org/wiki/Beta-binomial_distribution

The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution.
Imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn. Likewise, if a black ball is drawn, then two black balls are returned to the urn. If this is repeated n times, then the probability of observing k red balls follows a beta-binomial distribution with parameters n, α and β.

{$\lim_{n\to\infty} \frac{E(|S_n|)}{\sqrt n}= \sqrt{\frac {2}{\pi}}$} for the expected translation distance after n steps in a one-dimensional random walk, as with Pascal's triangle.

The asymptotic function for a two-dimensional random walk as the number of steps increases is given by a Rayleigh distribution. The probability distribution is a function of the radius from the origin and the step length is constant for each step. {$P(r) = \frac{2r}{N} e^{-r^2/N} $}

Gamma distribution	{$f(x;\alpha,\beta) = \frac{ x^{\alpha-1} e^{-\beta x} \beta^\alpha}{\Gamma(\alpha)}$} where {$ x > 0 \quad \alpha, \beta > 0$}.	The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and with respect to a 1/x base measure) for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).
Beta distribution	{$f(x;\alpha,\beta) = C x^{\alpha-1}(1-x)^{\beta-1}$} where {$0 ≤ x ≤ 1$}, {$α, β > 0$}	Models the behavior of random variables limited to intervals of finite length. Models the random behavior of percentages and proportions.
Student's t-distribution	{$f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{\!-\frac{\nu+1}{2}},\! $}	Estimating the mean of a normally distributed population in situations where the sample size is small and the population's standard deviation is unknown. The t-distribution has heavier tails than the normal distribution, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero.

Factorial moment If a random variable X has a Poisson distribution with parameter λ, then the factorial moments of X are {$\operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=\lambda ^{r}$}.

M/M/1 probability distribution

The M/M/1 probability distribution suggests that the universe is a simulation.
Hermite. Measured space (0,0). No cases where nothing happens. Time is the evolution of nothing happening. It is the observer's point of view.
Bound state: Service time = arrival time. Can have periodic wave behavior.
Charlier: Service time is 0.

M/M/1 Queue

An M/M/1 queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers in the system, including any currently in service.
- Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1.
- Service times have an exponential distribution with rate parameter μ in the M/M/1 queue, where 1/μ is the mean service time.
  - Exponential distribution The probability distribution of the time between events in a Poisson point process (in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless.
- A single server serves customers one at a time from the front of the queue, according to a first-come, first-served discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one.
Samuel Karlin, James McGregor. Linear Growth, Birth and Death Processes