Physical Basis of Distribution

Discrete:

• Binomial: model #“success” in $n$ trials when the trials are iid with success probability $p$ e.g. $n$ 个产品中有多少不合格
• Negative Binomial: model #trials needed to achieve $k$ ”success“ e.g. 需要检测多少个产品才能检测出 $k$ 个不合格的
• Geometric: model #trials until 1st “success”
• Possion: model #independent events that occur in a fixed time or space e.g. 一个小时内到访商店的顾客数量。

Continuous:

• Normal: models the distrubution of process that can be thought as the sum of a number of component process (from Central Limit Theorem)
• Weibull: model the time to failure for components, with increasing/constant/decreasing failure rate e.g. 常应用于描述电子元器件故障所需的时间，因为电子元器件的故障概率会随着自身的老化而上升(increasing failure rate)
• Exponential: model the time between independent events, or a process which is memoryless e.g. the time to failure for a system that has constant failure ratio over time (1) “memoryless” 类似于马尔可夫链，即任意时刻 $P(t\vert t-1)=P(t-1\vert t-2)$。例如假设公交车等待时间满足指数分布，那么 P(再等5分钟|上辆车刚走10分钟) = P(再等五分钟|上两车刚走30分钟)。这是因为 $P(T>15\vert T>10)=P(T>35\vert T>30)$ (2) Exponential is a special case of Weibull with constant rate
• Erlang: model the sum of $k$ exponential random variables (1) Erlang is a special case of gamma
• Gamma($\alpha, \beta$): an extremely flexible distribution used to model nonnegative random variables
• Beta($\alpha, \beta$): an extremely flexible distribution used to model bounded (fixed upper and lower limits) random variables
• Uniform: model complete uncertainty, since all outcomes are equally likely (suitable to worst-case analysis)
• Triangular: model a process when only the minimum, most likely and maximum values of the distribution are known e.g. the minimum, most likely and maximum inflation rate we will have this year