The difference between probability function p (x), probability distribution function f (x) and probability density function f (x)

Statement

The main content of this article is reproduced to bloggers Maldives Maldives A brief book of Probability function P(x)、 Probability distribution function F(x)、 Probability density function f(x)

1. Discrete variables and continuous variables

Before we get to the topic , Let's make a few concepts clear ：
Discrete variable （ Or take a limited number of variables ）： Values can be enumerated one by one , And the total is certain , Such as the number of dice （1 spot 、2 spot 、3 spot 、4 spot 、5 spot 、6 spot ）.
Continuous variable （ Or take an infinite number of variables ）： Values cannot be enumerated one by one , And the total number is uncertain , Such as all natural numbers （0、1、2、3……）.

Discrete variables take a certain value xi Probability P(xi) It's a certain value （ Although many times we don't know the value ）, namely P(xi)≠0： for example , Roll the dice once to appear 2 The probability of a point is P(2)=1/6.

A continuous variable takes a certain value xi Probability P(xi)=0： For continuous variables ,“ The probability of taking a specific value ” The statement of is meaningless , Because the probability of taking any single value is equal to 0, It can only be said “ The probability that the value falls within a certain interval ”, or “ The probability that the value falls in the neighborhood of a value ”, That is to say P(a<xi≤b), And can not say P(xi). Why is that ？ Let's look at the following example ：
   for example , Take any number from all natural numbers , Ask this number equals 5 What's the probability of ？ Take one from all natural numbers , Of course, it is possible to get 5 Of , But there are infinite natural numbers , So take 5 Is the probability that 1/∞, That is to say 0.
   Another example is throwing darts , Although it is possible to fall in the bull's-eye , But the probability is also 0（ Regardless of proficiency and other factors ）, Because there are countless points on the target , The probability of each point is the same , Therefore, the probability of falling on a specific point is 1/∞=0.

According to the previous example ： In continuous variables ： The probability of 0 The event of is possible , The probability of 1 The event of does not necessarily happen .

2. Probability distribution and probability function

A probability distribution : It means that all values and their corresponding probabilities are given （ Not without one ）, Only for discrete variables meaningful . for example ：

A probability distribution

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 probability distribution .

3. Probability distribution function and probability function

Probability distribution and probability function are only meaningful for discrete variables , How to describe continuous variables ？
The answer is “ Probability distribution function F(x)” and “ Probability density function f(x)”, Of course, these two It can also describe discrete variables .

Probability distribution function F(x)： Give the probability that the value is less than a certain value , 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 back ）.
Probability distribution function F(x) The nature of ：

Properties of probability distribution function

Probability distribution function F(x) The role of ： Here's the picture
（1） give x Fall in a certain range (a,b] The probability of the interior ：P(a<x≤b)=F(b)-F(a)
（2） according to F(x) Slope judgment of “ Interval probability ”P(A<x≤B) The change of （ In fact, it is the probability density function to be mentioned later f(x)）（ Particular attention ： Is a judgment “ Interval probability ”, namely x Fall in the (A,B] The probability of , instead of x The probability of taking a certain value , This is the essential difference between continuous variables and discrete variables ）
   An interval (A,B] Inside ,F(x) More inclined , Express x The probability of falling within this interval P(A<x≤B) The bigger it is . As shown in figure, (a,b] Within the interval F(x) The slope of is the largest , If the whole value range is divided into δx=b-a The intervals are equidistant , be x Fall in the (a,b] The probability within is the greatest . Why? ？ because P(A<x≤B) )=F(B)-F(A), Only in (a,b] In this interval （ namely A=a,B=b）F(B)-F(A) To the maximum , That is, the vertical red line segment in the figure is the longest .

Analysis of probability distribution function

Probability density function f(x)： Given that the variable falls at a certain value xi In the neighborhood （ Or in a certain interval ） How fast does the probability change , The value of the probability density function is not probability , It's the rate of change of probability , The area below the probability density function is the probability .

Probability density function definition

Properties of probability density function  The relationship between probability distribution function and probability density function

Be careful

Appointment

4. Probability of continuous variables 、 Probability distribution function 、 The relationship between probability density functions

Probability of continuous variables 、 Probability distribution function 、 The relationship between probability density functions （ Take the normal distribution as an example ） Here's the picture ：
   For a normal distribution ,x Fall in the u The probability nearby is the greatest , and F(x) Is the cumulative sum of probability , So in u near F(x) The incremental change is the fastest , namely F(x) The curve is in （u,F(u)） The tangent slope at this point is the largest , This slope is equal to f(u).x Fall in the a and b The probability between is F(b)-F(a)（ The small red line segment in the figure ）, In the probability density curve f(x) And ab The enclosed area S. As shown in the figure below ：

probability 、 Probability distribution function 、 The relationship between probability density functions

The probability density function is at a certain point a Value f(a) What is the physical meaning of ？
We know f(a) Express , Probability distribution function F(x) stay a The rate of change of the point ( Or derivative ); Its physical meaning is actually x Fall in the a Probability in the infinitesimal neighborhood near the point , But it doesn't fall on a Probability of a point （ As mentioned before , Continuous variable single point probability =0）, To describe in mathematical language is ：

Relationship