Chapter II random variables and their distributions

Content Summary:

I. DEFINITION of random variable

Set is a random test, the sample space, if for each sample point, there is a unique fixed number corresponding to it, then the real value function is called a random variable (denoted).

The concept and properties of distribution function

1. Definition of distribution function

Set is a random variable, which is called the real value function defined on the

is the distribution function of the random variable.

2. The properties of the distribution function

(1),

(2) Monotone not reducing:,

(3)

(4) Right continuity:.

Note: The above 4 properties are the necessary and sufficient conditions for a function to be a distribution function of a random variable. In different textbooks, the definition of a distribution function may vary, for example, by its nature.

(5)

Note: This property is a description of the statistical law of the distribution function on random variables.

Three, discrete type random variable

1. The definition of discrete random variable

If all possible values of random variables can be listed at most, the random variables are called discrete random variables.

2. Distribution law of discrete random variables

(1) Definition: All possible values of discrete random variables and the probability value when taking each value, called discrete random variable distribution law, expressed as

Or in a tabular representation:

X1 X2 ... xn ...

Pk

P1 P2 ... pn ...

or recorded as

~

(2) Nature:,

Note: This property is a sufficient and necessary condition for the distribution law of a discrete random variable.

which

Note: The common distribution Law describes the statistical law of discrete random variables.

3. distribution function of discrete random variables

=, it is the right continuous ladder-like function.

4. The common discrete type distribution

(1) Two point distribution (0-1 distribution): Its distribution law is

That

0 1

P

1–p P

(2) Two distribution

(Ⅰ) The source of the two distribution-Chembonoulli test: The set is a randomized trial, with only two possible outcomes and, will be repeated independently, then called this series of repeated independent tests for the Bernoulli test.

(Ⅱ) Definition of distribution of two items

To indicate the number of events occurring in the re-Bernoulli test, the distribution law of the random variable is

， ，

The two-item distribution of the random variable subject to the parameter is recorded.

Note: That is two point distribution.

(3) Poisson distribution: If the distribution law of the random variable is

， ，

The Poisson distribution of the random variable following the parameter is called (or.

Iv. Continuous random variables

1. Definition of continuous random variable

In the case of random variables, there is a nonnegative function defined on it, so that for any real number, there is always a constant random variable, which is called the probability density function, the probability density, for the sake of clarity, sometimes written as.

2. The properties of probability density function

(1)

Note: This property is a necessary and sufficient condition for the probability density of a continuous random variable.

(2) for continuous random variable, must be consecutive, but not necessarily continuous, at the point of continuous, there,

(3) to arbitrary real numbers thereby to arbitrary real numbers, which have

。

Note: The common probability density describes the statistical law of continuous random variables.

4. Common Continuous-type distributions

(1) Uniform distribution

To represent the placement coordinates in a geometric pattern, the distribution function is

，

Its probability density is

，

The uniform distribution on the interval of the said obedience is recorded as.

(2) Exponential distribution

If the probability density of the random variable is

，

The distribution function of the exponential distribution which is called the parameter is

。

(3) Normal distribution

(Ⅰ) Standard normal distribution: If the probability density of the random variable is

，，

It is said to obey the standard normal distribution, and the distribution function is

，

(Ⅱ) General distribution: If the probability density of the random variable is

，，

Then the normal distribution of the parameter is called, and its distribution function is

，

(Ⅲ) The nature of the normal distribution:

Satisfying symmetry, i.e.,;

If, then, that is, thereby having;

Note: By the above properties, the calculation of normal distribution can be converted to the calculation of the standard normal distribution, and for the standard normal distribution of the distribution function value, there are tables available, according to the symmetry, at that time, according to the calculated value.

If, then

(Ⅳ) The upper-division point of the Standard normal distribution: set, for the given, the point to be satisfied is the upper point of the standard normal distribution.

function distribution of random variables

1. function distribution of discrete random variables

The distribution law of the discrete random variable is

is a continuous function), the distribution law is

Case one: When all is not the same, the distribution law is

Case two: If you know a certain, then there are

，

In general, the distribution law is

，

2. function distributions of continuous random variables

The probability density of a continuous type of random variable is

Case one: If the function is everywhere, it is also a continuous random variable, whose probability density is

where = yes inverse function.

Case two: If the function is not strictly monotonous, the probability density can be calculated in two steps:

The first step is to find the distribution function,

The second step is the derivation number.

Six, some notes

1. If there are undetermined constants in the distribution function, the determination of the constant is the nature of the utilization: or.

2. If there are undetermined constants in the probability density function (distribution law), the determination of the constant is the property of the use (Distribution Law): (;

3. If the continuous type of random variable, to arbitrary real numbers;

4. In the distribution law of discrete random variables, two elements are indispensable, that is, all possible values and the probability value of each value, and the distribution function of discrete random variable is the piecewise function of right continuous and ladder-like.

5. If the continuous type of random variable, according to mutual demand.

Basic requirements

1. Master the concepts of random variable, discrete random variable, continuous random variable, distribution function, distribution law and probability density function, and understand the properties of distribution function, distribution law and probability density function.

2. The distribution function, distribution law and probability density function of random variable can be obtained by using random variable to describe the event, and the distribution of the random variable function is obtained.

3. Proficient in six kinds of commonly used distribution;

4. The distribution function is obtained by the distribution law or probability density function, known distribution law or probability density function.

Key content

The concept of random variable, distribution function, distribution law and probability density function, the calculation of distribution function and probability density function, the distribution of random variable function.

Analysis of typical examples

Example 1 the Distribution law and distribution function of random variables are obtained by means of the maximum number of the ball in a box with 5 balls labeled 1,2,3,4,5, from which 3 can be taken.

Analysis: In this case, all possible values are 3,4,5, and the probability of taking each value (event) is the probability of classical probabilistic type, and then the distribution function is derived according to the relation of distribution law and distribution function.

Solution: All possible values are 3,4,5,

At that time, the number was taken out (1,2,3),

At that time, the number was taken out (1,2,4), (1,3,4), (2,3,4),

At that time, the number was taken out (1,2,5), (1,3,5), (1,4,5), (2,3,5), (2,4,5), (3,4,5),,

So the distribution law is

X

3 4 5

Pk

1/10 3/10 3/5

The distribution function derived from the formula can be

Example 21 batches of 9 authentic 3 defective, from any one, if removed defective no longer put back, for the removal of genuine defective number before the distribution law.

Analysis: In this case, all possible values are 0,1,2,3, and the probability of taking each value (event) is a classical probability.

Solution: All possible values are 0,1,2,3, which means that the first take out is genuine, then the distribution law of the multiplication formula is

Similarly

Example 31 target is a two-metre radius disk, the probability of hitting the target on a concentric disk is proportional to the area of the disc, and set the shooting can be in the target, to indicate the distance between the impact point and the center, the distribution function.

Analysis: According to the definition of distribution function.

Solution: Set the distribution function is, if, it is impossible event, at this time,;

If by the order, to determine the constant, take, then there, and, therefore, thus

；

If, then is the inevitable event, therefore;

Sum up

Example 4 set the distribution function of random variable to

Try to determine the constants and ask.

Analysis: According to the preceding note, the application of the right continuity can be used to find the constant, and then apply the nature of the random variable in the description of the statistical law to find the probability.

Solution: By the right continuity of the distribution function, the relationship between the probability and the distribution function

Note: From this example, it can be seen that the distribution function is neither continuous nor ladder-like, which shows that there are not discrete and discontinuous random variables.

Example 5 set the distribution function of random variable as, and

(1) coefficient, (2) The probability of Falling, (3) the probability density.

Analysis: Based on the property and the probability density function of the relationship between the solution.

Solution: (1) because, know

, the solution,

So.

(2)

=

(3),

Example 6, the probability density of random variable is set,

(1) coefficient, (2) (3) the distribution function.

Analysis: The relationship between the distribution function and the probability density function is solved according to the properties.

Solution: (1) because, that is,

(2),

(3)

Was

At that time, +,

So the distribution function.

Example 7 The life of a TV set (in years), with the following probability density function

(1) The probability of a TV set to live for up to 6 years,

(2) The probability that the life expectancy is between 5-10 years,

Analysis: It is known that the probability density function of the continuous random variable is the probability, according to the previous formula.

The life of a television is recorded as

(1)

(2)

Example 8 A bus is passed every 5 minutes, passengers arrive at the station at any one time is possible, please

(1) The probability that the passenger waiting time should not exceed 3 minutes,

(2) Passengers to wait 3 times a week, if more than 3 minutes to leave, used to indicate the passenger in a week waiting for the number of buses, the distribution law, and ask.

Analysis: By the question, the passenger waiting time should obey the uniform distribution, asks the passenger waiting time not exceeding 3 minutes probability, is according to the probability density to ask for the probability, also from the question to know, is obeys two item distribution the random variable.

Solution: (1) to indicate the waiting time, from the subject to be known to obey the uniform distribution on [0,5], the probability density of

，

The probability of a passenger waiting time not exceeding 3 minutes;

(2) by the need to be, and thus

= =.

Example 9 to set the random variable, ask:

(1)