Recently, I'm looking at some knowledge of probability and statistics , By the way, I made some notes .
Basic probability model
Here are three concepts , Classical probability , Frequency school , Bayesian school .
In this model , All possible results of randomized experiments are limited , And the probability of each basic result is the same
such as ： Toss an even coin , There are only two results （ Suppose the coin doesn't stand up ）, Face up and back up , So the probability of facing up is 0.5. This is a calculation based on the classical probability model .
It is considered that the parameter to be estimated is an unknown constant , Through many tests , The ratio of the number of events to the total test was counted , Get the value of the parameter to be estimated .
such as ： Estimate the probability of getting a positive by throwing an even coin . We do it 1000 Tests , Yes 498 Second up , So the probability of getting positive is 0.498.
* Bayesian school
The parameter to be estimated is not a fixed constant , It's a random variable （ Obey a certain distribution ）. About this random variable , We can have a priori estimation of its distribution according to common sense or other objective facts （ belief ）, Then adjust the distribution according to the experiment , Finally, the posterior distribution of the random variable is obtained .
This idea solves the problem of test deviation caused by too few experiments in frequency school experiment , such as , Toss a homogeneous coin 5 second , this 5 It's always face up , According to the viewpoint of frequency school , What is the probability of coin toss face up
P( Face up )=55=1, This is obviously out of the ordinary sense .
Now define the event A=( Toss the coin face up once ),B=( throw 5 Sub coin ,5 Second up ). In the framework of Bayes , According to common sense, we think that the probability of tossing a coin face up is 0.5, So we can assume that the prior obeying parameter is
Beta(10,10) The distribution of , Then according to Bayesian theorem P(A|B)=P(A)P(B|A)P(B) Can be calculated in the event B Under the condition of occurrence A The probability distribution of is distribution Beta(15,10)
, The expected value of this distribution is 0.6. Through Bayesian framework , We calculated that the probability of the coin facing up is still close 0.5 Value of , More in line with our common sense .（ about Beta The detailed calculation of distribution and posterior probability will be introduced in later chapters ）
This graph is a prior distribution plotted separately Beta(10,10)（ blue ） And posterior distribution Beta(15,10)（ green ）
Conditional probability and mutual independence
conditional probability , if P(B)>0, be P(A|B)=P(AB)P(B) Recorded as an event B In case of occurrence ,A Probability of occurrence .
If P(A|B)=P(A), be A And B Independent of each other and ,P(A∩B)=P(A)P(B)
Common distribution of unit random variables
* Bernoulli distribution （0-1 distribution ） Bernoulli
The probability distribution is
Binomial distribution binomial
to the full n Independent Bernoulli test .N In the second independent test , The incident happened K Probability distribution of times
uniform distribution Uniform
Go to the room a,b Between the probability density function of uniform distribution
* exponential distribution
The parameter is λ The probability density of exponential function
* Normal fraction
The mean value is μ, The standard deviation is σ Probability density of normal distribution