Notes on Probability Theory and Mathematical Statistics (2) Uncle Gao Mathematics, there is one month of urgent review at the end of the term

Notes on Probability Theory and Mathematical Statistics (2)

4. Conditional probability and multiplication formula

P（B|A)= P（AB）/ P（A）

Properties of Conditional Probability

The memory method of the above formulas
(1) The conditional probability is also a probability
(2) The conditional probability satisfies the normative
(3) (2) variant, after the negation of S=A+A, the 3 formula
(4) conditions can be obtained Probability satisfies the probability addition formula

5. Total probability formula and Bayesian formula

6. Independence of Events

1. Independence of events
   If A and B are two events, if P(AB)=P(A)P(B) is satisfied, then events A and B are said to be independent of each other
2. Related conclusions about event independence
   
   

7. Discrete random variable

1. Random variable
   Definition: A random variable is a single real-valued function defined on the sample space of a random experiment, denoted as X=X(e) Benefits:
   Random events are mapped to a set of real numbers
   Notes:
   (1) Random variable X is a simple real The value function, that is, the result of the random experiment has only one value corresponding to it on the real number axis. Of course, the real values ​​corresponding to different elements may be the same (2) X(e) reflects the description of random events
   (
   3 ) Each value of X(e) has a certain probability
2. discrete random variable

• All values ​​of discrete random variables are finite or infinite. The infinite number that can be listed refers to the one-to-one correspondence with the natural numbers.
• The distribution law of a discrete random variable: Assuming that all possible values ​​of a discrete random variable X are x, the probability of X taking each possible value is called the probability distribution of the random variable X, also called the distribution law.

3. Common Discrete Random Variables
   1. 0-1 distribution, there are only two results of random experiments, and the random variable X has only two possible values ​​of 0 and 1, and its distribution law can be written as p(0)=p,p(1)=1-p
   2. Binomial distribution A
      random experiment with only two outcomes is called a Bernoulli experiment, and this experiment is repeated n times independently, called an n-fold Bernoulli experiment. The n-fold Bernoulli test fits the binomial distribution.
      
      We call the distribution that the random variable X of the number of occurrences of A in the n-fold Bernoulli experiment obeys is called the binomial distribution. denoted as B(n,p)
   3. Poisson distribution
      The Poisson distribution is suitable for describing the number of random events occurring per unit time (space), such as the number of vehicles arriving at a gas station within one hour.
      
   4. Poisson's theorem
      The limit of the binomial distribution obeys the Poisson distribution, λ=np
      
4. The distribution function of random variables The
   distribution function means adding the probability corresponding to all the values ​​​​less than or equal to x in the value of the random variable X, and the definition domain is R
   
   F(x) is a non-decreasing function
   P(a<x<=b) =F(b)-F(a)
   F(x) range is between 0-1

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