Chapter three multidimensional random variables and their distributions

Content Summary:

One or two-D random variable

1, the definition of two-dimensional random variable: Set E is a random test, its sample space is defined on the s of the random variable, it is called two-dimensional random vector or two-dimensional random variable.

2, two-dimensional random variable distribution function definition: Set is a two-dimensional random variable, for any real number, two-element function:

A distribution function called a two-dimensional random variable, or a joint distribution of two dimensional random variables.

3. The properties of the distribution function of two dimensional random variables:

(1) is a monotone and non descending function of a variable.

(2), and for arbitrarily fixed, for any fixed,.

(3) about the variable and the separate right continuous

(4) for any

It is important to note that as long as the two-element function that satisfies these four bars must be a distribution function.

4. The distribution law of two dimensional discrete random variables, or the joint distribution of random variables, is called a finite pair or an infinite set of two-dimensional random variables.

The distribution function of two-dimensional discrete random variables is:

5, for a two-dimensional random variable distribution function, if there is a nonnegative integrable function for any real number has

is called a continuous two-dimensional random variable, a function called the probability density of a two-dimensional random variable, or a joint probability density called a random variable.

6. The properties of the probability density of two-dimensional continuous random variables:

(1); (2);

(3) setting is the area on the surface, the probability of the point falling inside is

；

(4) If in continuous, then.

Note: Similar to two-dimensional random variables, we can define multidimensional random variables and their distribution functions, and discuss the properties of distribution functions.

Ii. condition distribution and edge distribution

1, two-dimensional random variables as a whole, with distribution functions, and x and y as random variables, distribution functions are recorded as and, in turn, the two-dimensional random variables about x and the y of the edge distribution function, and.

(1) Two-dimensional discrete random variable, distribution law, then the distribution of x and Y are respectively

And the law of fringe distribution, which is called about and respectively.

On the fixed, if it is said

is the condition distribution law of the random variable under the condition.

On the fixed, if it is said

is the condition distribution law of the random variable under the condition.

(2) Two-dimensional continuous type random variable, the probability density is, and the probability density is respectively

and respectively called about X and the edge probability density on Y.

On the fixed, if it is said

is the conditional probability density of the random variable under the condition.

is the conditional distribution function of the random variable under the condition.

On the fixed, if it is said

is the conditional probability density of the random variable under the condition.

is the conditional distribution function of the random variable under the condition.

Note: The concepts of joint distribution functions, edge distributions, and conditional distributions can be similar to those extended to dimensional random variables.

The independence of random variables

1, the distribution function of two-dimensional random variable is, the edge distribution function is and, if for any real number, there are

The random variables and the said are independent of each other.

When it is a discrete random variable, the necessary and sufficient conditions are independent of each other for all possible values

When it is a continuous type of random variable, the necessary and sufficient condition of independence is the equation

was almost everywhere.

Note: The so-called "almost everywhere" refers to the plane to remove the area of zero outside, the establishment everywhere.

2, set is the dimensional random variable, its distribution function is

Which is any real number.

If the distribution function of the dimensional random variable is known, then the dimension edge distribution function is determined. For example, the distribution function of a dimensional random variable is

If the probability density of the dimensional continuous type random variable is, then the dimension edge probability density is determined. For example, the probability density of a dimensional random variable is

If you have any

Are said to be independent of each other.

if for any;

Are said to be independent of each other.

If they are independent of each other, they are independent of each other, which is a continuous function.

Distribution of functions of four or more random variables

1, set is a discrete type of random variable, distribution law is

，

The Distribution law is

is a continuous random variable, the density function is, the distribution function is

2, set is a continuous type of random variable, the density function is, the distribution function is

In particular, when and mutually independent, distribution densities are and, respectively, the distribution functions for

The following conclusions can be obtained:

If, and independent of each other, then

If, and independent of each other, then

3, set discrete random variable, and the independent, distribution law is the same, the distribution law is

4, set is a mutually independent random variable, their distribution function is, remember, the distribution function is

The distribution function is

In particular, when there is the same distribution function,

Analysis of typical examples

Example 1, the joint probability density function of two dimensional random variables is

wherein, the two-dimensional normal distribution, which is called the obeying parameter, is recorded as.

The edge distribution and condition distribution are all normal distributions, and

。

The conditions for a given real number are distributed as

The conditions under which the given real numbers are distributed are

Solution: With probability density

Because So,

That

Similarly

That

The conditional probability density function for a given real number is

That

The conditional probability density function for a given real number is

That

Therefore, the boundary distribution and condition distribution of two-dimensional normal variables are normal distribution.

Example 2, set the random variable with probability density of

(1) Finding and determining whether the random variables are independent of each other;

(2) The probability density function is obtained.

Solution: (1)

Similarly

Because

Random variables are not independent of each other.

(2) The probability density function is

Example 3, set random variable with probability density

The probability density function is obtained.

Solution: Set the distribution function for,

At that time,;

Was

So the probability density function

Example 4, set the random variable independent, with the same probability density

The probability density function is obtained.

Solution:

The distribution function is

So the probability density function is

The distribution function is

So the probability density function is

Example 5, if, and mutually independent, then

Prove:

So.

Because, and independently of each other,

Here we use mathematical induction to prove.

When the conclusion is established. Suppose there is,.

They are independent of each other, and, therefore, have

。

Example 6, the probability density function of random variable is

Proven and not mutually independent, and mutually independent.

Solution:

Similarly

Because

So and not independent of each other.

Was

Was

；

Similarly, at that time,;

Evidently, when;

At that time,;

That

and

Similarly

So, arbitrarily for real numbers, there are, then and mutually independent.

Self-Test questions

First, fill in the blanks (3 points per empty, total 21 points)

(1) Random variables are independent of each other, and the distribution law of random variables is

(2) Set and are two random variables,,, then.

(3) Known random variable has probability density

Then, the edge density function.

(4) Known random variable has probability density

Then, =.

(5) Set up mutually independent, and,

The

Two. (9) Set random variable with probability density of

The probability density function is obtained.

Three (10 points). Random variables and independent, their probability density functions are

The probability density function is obtained.

Four, (10) Set random variables and mutual independence,

Five, (10) Set random variable with probability density of

(1) To find the edge probability density function, (2) to determine whether the random variable is independent of each other;

(3) The conditional probability density function is obtained.

Six, (10) Set random variables independent of each other, with the same probability density

The probability density function is obtained.

Seven, (10) random variables and independent of each other, their probability density functions are respectively

The probability density function is obtained.

Eight, (10 points) set random variable