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1. Write in front
Before entering the topic, let's clarify a few concepts:
Discrete variables (or variables with a limited number of values) : the values can be listed one by one, and the total number is determined, such as the number of points that appear in the dice (1 point, 2 points, 3 points, 4 points, 5 points, 6 points point).
Continuous variables (or variables with an infinite number of values) : the values cannot be listed one by one, and the total number is uncertain, such as all natural numbers (0, 1, 2, 3...).
The probability P(xi) of a discrete variable taking a certain value xi is a definite value (although we often don’t know what the value is), that is, P(xi)≠0: For example, the probability of throwing a dice and appearing 2 points is P(2)=1/6.
The probability of a continuous variable taking a certain value xi P(xi)=0 : For continuous variables, the statement "the probability of taking a specific value" is meaningless, because the probability of taking any single value is equal to 0 , can only say "the probability that the value falls within a certain interval" , or "the probability that the value falls within the neighborhood of a certain value" , that is, we can only say P(a<xi≤b), not P( xi). why is it like this? And look at the following example:
For example, if a number is randomly selected from all natural numbers, what is the probability that this number is equal to 5? Taking one of all natural numbers, it is of course possible to get 5, but there are infinitely many natural numbers, so the probability of getting 5 is 1/∞, which is 0.
Another example is throwing a dart. Although it is possible to land on the bullseye, its probability is 0 (regardless of other factors such as proficiency), because there are countless points on the target board, and the probability of each point is the same, so it falls on the bullseye. The probability of a specific point is 1/∞=0.
According to the previous example, it can be seen that in continuous variables: events with a probability of 0 are possible, but events with a probability of 1 may not necessarily occur. (PS: You can understand it from the definition of the probability formula of continuous random variables)
2. Enter the theme
Probability distribution : All values and their corresponding probabilities are given (one less is not enough), which is only meaningful for discrete variables . For example:
Probability distributions
Probability function : The probability of occurrence of each value is given in functional form, P(x) (x=x1, x2, x3,...), only meaningful for discrete variables , in fact it is a mathematical description of the probability distribution .
Probability distributions and probability functions are only meaningful for discrete variables, so how to describe continuous variables?
The answer is " probability distribution function F(x) " and " probability density function f(x) ". Of course, these two can also describe discrete variables.
Probability distribution function F(x): gives the probability that the value is less than a certain value, which is the cumulative form of probability , namely:
F(xi)=P(x<xi)=sum(P(x1),P(x2),...,P(xi)) (for discrete variables) or integral (for continuous variables, see the figure below ).
Properties of the probability distribution function F(x) :
Probability Distribution Function Properties
The role of the probability distribution function F(x) : as shown below
(1) Give the probability that x falls within a certain interval (a,b] : P(a<x≤b)=F(b)-F(a)
(2) Judging the change of "interval probability" P(A<x≤B) according to the slope of F(x) (actually the probability density function f(x) to be mentioned later) (special attention: it is to judge the "interval probability" ", that is, the probability that x falls in (A,B], not the probability that x takes a certain value, which is the essential difference between continuous variables and discrete variables)
In a certain interval (A,B], the more inclined F(x) is, the greater the probability P(A<x≤B) that x falls in the interval. As shown in the figure, F(x) in the interval (a,b) The slope is the largest. If the entire value interval is equally spaced at the interval of δx=ba, then the probability of x falling in (a,b] is the largest. Why? Because P(A<x≤B) )=F(B )-F(A), in all intervals, only in the interval (a,b] (that is, A=a, B=b) F(B)-F(A) reaches the maximum value, that is, the vertical red in the figure The longest line segment.
Analysis of Probability Distribution Function
Probability density function f(x) : It gives the probability that the variable falls within the neighborhood of a certain value xi (or within a certain interval) changes quickly . The value of the probability density function is not the probability, but the rate of change of the probability. The probability density function is below The area of is the probability .
Probability Density Function Definition
Probability Density Function Properties
The relationship between probability distribution function and probability density function
The relationship between the probability, probability distribution function, and probability density function of continuous variables (taking the normal distribution as an example) is as follows:
对于正态分布而言，x落在u附近的概率最大，而F(x)是概率的累加和，因此在u附近F(x)的递增变化最快，即F(x)曲线在（u，F(u)）这一点的切线的斜率最大，这个斜率就等于f(u)。x落在a和b之间的概率为F(b)-F(a)（图中的红色小线段），而在概率密度曲线中则是f(x)与ab围成的面积S。如下图所示：
概率、概率分布函数、概率密度函数之间的关系
三、物理意义
概率密度函数在某点a的值f(a)的物理意义到底是什么？
我们知道f(a)表示，概率分布函数F(x)在a点的变化率(或导数)；其物理意义实际上就是x落在a点附近的无穷小邻域内的概率，但不是落在a点的概率（前已述及，连续变量单点概率=0），用数学语言描述就是：
关系
原作者：简书——马尔代夫Maldives，点击名字进入原文查看。
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