Definition 1: Suppose the sample space of experiment E is S={e}, and X=X(e) and Y=Y(e) are two random variables defined on S. The vector (X, Y) composed of these two random variables is called a two-dimensional random variable or a two-dimensional random vector.

Definition 2: Let (X, Y) be a two-dimensional random variable. For any real number x, y, the binary function is called the distribution function of the two-dimensional random variable or the joint distribution function of the random variables X and Y.

Definition 3: Suppose the sample space of test E is S={e}, and Xi=Xi(e) is a random variable defined on S, i=1,2,…,n, composed of these n random variables An ordered group of random variables is called an n-dimensional random variable or random vector. Let be an n-dimensional random variable. For any real number , the n-ary function is called the distribution function of the n-dimensional random variable or the joint distribution function of n random variables.

Properties of distribution function F(x,y):

Definition: If the values ​​of the two-dimensional random variable (X, Y) are finite pairs or listable pairs, then (X, Y) is said to be a discrete random variable.

It is called the (probability) distribution law of two-dimensional discrete random variables (X, Y), Or called the joint (probability) distribution law of X and Y.

The basic properties of the distribution law of two-dimensional discrete random variables (X,Y):

Theorem: Suppose the distribution law of (X, Y) is the probability that a random point (X, Y) falls in any area D on the plane. The sum is the sum of all i, j such that (xi, yj)D .

Especially

Definition: Suppose the distribution function of a two-dimensional random variable (X, Y) is F (x, y). If there is a non-negative integrable function f (x, y) such that for any real number x, y, it is always said that ( X, Y) is a two-dimensional continuous random variable, and the function f(x, y) is called the probability density of the two-dimensional continuous random variable (X, Y), or the joint probability density of the random variables X and Y.

The probability density f(x,y) of (X,Y) has the following basic properties:

On the contrary, if the binary function f(x,y) satisfies the above two basic properties, then it must be the probability density of a certain two-dimensional random variable (X,Y).

If the probability density f(x,y) is continuous at point (x,y), then we have

Calculate probability using probability density

Commonly used two-dimensional continuous random variables

Distribution function of component X: Call FX(x) the edge distribution function of (X,Y) with respect to X;

Distribution function of component Y: FY(y) is called the marginal distribution function of (X,Y) with respect to Y.

Theorem: The distribution law of two-dimensional discrete random variables (X, Y) is , then the edge distribution law of (X, Y) with respect to X The edge distribution law of (X, Y) with respect to Y is

The marginal distribution law of (X,Y) with respect to

Under the condition that one component is known to take a certain value, the distribution law of the other component is called the conditional distribution law.

definition:

Suppose the probability density of the two-dimensional continuous random variable (X, Y) is f (x, y), then the probability density of the component X is recorded as fX (x), which is called the edge probability density of (X, Y) with respect to X; The probability density of component Y is recorded as fY(y), which is called the edge probability density of (X,Y) with respect to Y.

Suppose the probability density of two-dimensional continuous random variable (X, Y) is f (x, y), then

definition:

Calculation formula

theorem

Discriminant theorem

definition