参考; wikipedia / Probability_distribution

Ⅰ概率分布 （probability distribution）

在probability theory 和 statistic（统计学）中，probability distribution 就是 mathematical function。probability distribution 提供了experience 中different 可能output发生的概率（possibility）。

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

如果random variable X用于表示抛硬币的outcome（“实验”），那么X的probability distribution将取X为X =头部的value 为5，而0.5代表X =尾部 （假设硬币是 fair）。random phenomena的示例可以包括experiment或survey的results。

For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails (assuming the coin is fair). Examples of random phenomena can include the results of an experiment or survey.

probability distribution是根据基础sample space 指定的，该simple space是观察到的随机现象的所有可能结果的set。sample space可以是实数集或一组向量，或者它可以是non numerical value 的list; 例如，coin flip的sample space将是{head，tails}。

A probability distribution is specified in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be {heads, tails}

probability distribution 按照random variable的不同分为discrete probability distribution 和 continuous probability distribution 两种。

discrete probability distribution can be describe by probability mass function下图说的很清楚，probability mass function对应的冲激值就是probability value(概率值)。

continuous probability distribution can be describe by probability density function在这幅图上，any individual outcome 的probability 都是0

Probability distributions are generally divided into two classes. A discrete probability distribution (applicable to the scenarios where the set of possible outcomes is discrete, such as a coin toss or a roll of dice) can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function. On the other hand, a continuous probability distribution (applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

Ⅱ Instruction

为了定义最简单情况的probability distribution，有必要区分discrete和continuous random variable。在discrete情况下，指定(specify)probability mass function就足够了p为每个可能的结果分配probability：例如，当投掷一个公平的骰子时，六个值1到6中的每一个具有1/6的probability。然后将event的probability定义为满足事件的结果的probability的sum; 例如，event “骰子滚动even value”的probability是
${\ p（2）+ p（4）+ p（6）= 1/6 + 1/6 + 1/6 = 1/2。}$ p（2）+p（4）+p（6）=1/6+1/6+1/6=1/2。
相反，当random variable通常从continuum体获取值时，任何单个结果的probability为zero，并且只有包含无限多个结果的事件（例如间隔）才能具有positive probability。例如，给定物体正好重500克的probability为零，因为随着我们的测量仪器的精度增加，精确测量500克的probability趋于零。然而，在quality control中，可能要求包含490g和510g之间的“500g”包装的probability应不小于98％，并且这种需求对测量仪器的准确性不太敏感。

continuous probability distribution可以用几种方式描述。 probability density distribution描述了 the infinitesimal probability(无穷小的概率) of any given value,，并且该结果位于一个给定的interval的probability可以通过计算 integrating的probability density function 在所述间隔。另一方面， cumulative distribution function 描述 random variable不大于给定值的概率; 结果位于给定区间的概率可以通过获取区间的端点处的累积分布函数的值之间的差来计算。cumulative distribution function是probability density function 的原函数 (antiderivative)

也就是说：
Discrete probability distribution function p.m.f. ( Probability mass function) ,random variable 对应的value 就是probability.
Continuous probability distribution function: p.d.f,Probability density function:
given any random variable 的value, probability都是无穷小或者为0，我们要么就用p.d.f来describe given interval 的probability. 要么就换另一种 cumulative distribution function describes the probability that the random variable is no larger than不大于 a given value。

Discrete probability distribution也可以用cumulative distribution function

Ⅲ some terms

参见： https://en.wikipedia.org/wiki/Probability_distribution