Basic mathematical knowledge (update)

One 、 Probability distribution function and probability density function of continuous variables

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)., Look at the example below ：                                        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.

Probability distribution function F(x) and Probability density function f(x) Used to describe continuous variables                                                                     Definition of probability distribution function ： 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)) Discrete variables                                                                                  = Find integral Continuous variable                                                                                         Probability distribution function F(x) The role of ：（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)）, 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 the figure , In the interval a,b Inside F(x) The slope of is the largest , That is to say x Fall in the range a,b The probability of the interior P(a<x≤b) The bigger it is , Its value is equal to  F(b)-F(a).

  Probability density function definition ： 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 .

  Properties of probability density function ：

The relationship between probability distribution function and probability density function is shown in the figure below ：

  Be careful ：

  Probability of continuous variables 、 Probability distribution function 、 The relationship between probability density functions is shown in the figure below ：

  My summary ： For the abscissa xi,F(x) The ordinate of indicates that the value is less than xi Probability ,f(x) The ordinate of represents the value at xi Nearby probability .

Two 、 A priori probability and a posteriori probability