Suppose the sample space of random experiment E is S={e}. If for each sample point, there is a certain real value X(e) corresponding to it, then the function X=X(e) whose value is a real number is called Random variable, abbreviated as X.

Let X be a random variable. For any real number x, let F(x) be the probability distribution function of the random variable X, referred to as the distribution function. recorded as

(1) Value range: , and

(2) Monotonic and non-decreasing: that is

(3) Right continuous,

(4)

(5)

Describe the probability rules of the values ​​of discrete random variables.

Has the same effect as the distribution function,

It is more direct and simple to describe the probability rules of random variable values ​​than distribution functions.

(1)Formula method:

(2) List method or matrix method:

(1)

(2)

(1) Distribution function of X

(2) For any interval I, there is

(3) The distribution law can be determined by the distribution function

definition

The binomial distribution is derived from the n-fold Bernoulli test.

distribution law

Definition of binomial distribution

The principle of small probability has two sides: a small probability event is almost impossible to occur in one experiment; a small probability event is almost certain to occur in many experiments.

definition

Poisson's theorem

When n is large and p is small, the binomial distribution has an approximate relationship with the Poisson distribution. Generally, when n>=10 and p<=0.1, there is an approximate formula:

Poisson distribution lookup table:

Suppose there are M pieces of genuine products and N pieces of defective products in a batch of products. If n pieces are randomly selected from them, the number of defective products X is a discrete random variable, and its probability distribution is: . This distribution is called hypergeometric distribution.

Suppose the distribution function of random variable Random variables are called function f(x) as the probability density function (or distribution density function) of random variable X, or probability density for short.

(1) For all x∈(-∞, ∞), f(x)≥0,

(2)

On the contrary, any integrable function f(x) in the real number field with properties (1) and (2) can become the probability density function of a continuous random variable.

Suppose X is a continuous random variable, the distribution function is F(x), and the probability density is f(x), then we have

If a continuous random variable X, its probability density function is: Then X is said to obey a uniform distribution on the interval [a, b]. Record it as X~U[a,b].

If the probability density of a random variable

definition

The probability density curve of the normal distribution has the following properties:

The normal distribution with parameters μ=0, σ=1, that is, N(0,1), is called the standard normal distribution. Its probability density and distribution function are represented by φ(x) and φ(x) respectively, that is, we have

subtopic

Φ(x) can be looked up in the table

The lower α quantile of the standard normal distribution N(0,1)

The relationship between the general normal distribution and the standard normal distribution N(0,1)