Author: Mathematics Department of Tongji University
/ 1203.html
reading comprehension
the first paragraph
Random variables and their distribution
First, the definition of random variables
Many test results in random experiments are expressed by quantity. For example,
(1) How many points will appear when you roll an even dice? x？ The value of;
(2) How many times will each insured vehicle claim compensation each year? n？ , the amount of each claim? y？ , here? n？ And then what? y？ The value of;
(3) Random error of measurement? ε? The value.
There are still many test results in random tests that are not expressed in quantity. At this point, you can set variables as needed, for example,
(1) Throw a uniform coin and observe its upward side. What is the sample space? Ω? = {Head up, tail up}. In this case, you can set a variable as follows? x？ ：
Here it is. x？ The value of corresponds to the following random events:
{? x？ = 1}={ face up}，{？ x？ =0}={ tails up}.
(2) Throw three even-numbered coins and observe the upward side. What is the sample space? Ω? ={? HHH？ ,? HHT？ ,? HTH？ ,? THH？ ,? HTT？ ,? THT？ ,? TTH？ ,? TTT？ }, of which? h？ It means face up. t？ It means face up. What if a variable at this time? x？ It means "the number of times the heads are up in three throws", then? x？ And the values of the sampling points have the following correspondence:
Here it is. x？ The value of corresponds to the following random events:
{? x？ =0}={ mantissa goes up 0 times} = {? HHH？ },{? x？ = 1}={ tail up 1 time} = {? HHT？ ,? HTH？ ,? THH？ },{? x？ =2}={ Tail up twice} = {? HTT？ ,? THT？ ,? TTH？ },{? x？ =3}={ Tail up 3 times} = {? TTT？ }。
Below, we give a general definition of random variables.
Definition 1? In a randomized trial? e？ Yes,? Ω? Is the corresponding sample space, if correct? Ω? Is every sample point there? ω? There is only one real number? x？ (? ω? ), then take this domain as? Ω? Single-valued real function? X=X？ (? ω? ) What's your name? (one-dimensional) random variable? .
Random variables generally use capital letters? x？ ,? y？ By analogy, the values of random variables are generally expressed in lowercase letters? x？ ,? y？ If a random variable can only take finite values, it is called a discrete random variable. If the value of a random variable is full of an interval (or the union of several intervals) on the number axis, it is called a non-discrete random variable. Continuous random variables are the most common types of non-discrete random variables.
The definition of random variable can be intuitively interpreted as: random variable? x？ Is a function of the sample points. The independent variable of this function is the sample point, which may or may not be a number. The domain is a sample space, and the dependent variable must be a real number. This function can make different sampling points correspond to different real numbers, and can also make multiple sampling points correspond to a real number.
The introduction of random variables is a sign of the development of probability theory, which makes up for the defect that there are many kinds of random events in random experiments and it is difficult to summarize their probability laws, because if the distribution of random variables is known, the probability of any random event in random experiments can also be obtained. In addition, after introducing random variables, we can use calculus tools in mathematics to discuss the distribution of random variables.
Second, the distribution function of random variables
Random variable? x？ Is it a sample point? ω? A real function, in order to master? x？ Statistical laws, we need to know? x？ The probability of taking a value in an interval. because
{? Answer? & lt？ x？ ≤? b？ }={? x？ ≤? b？ }-{? x？ ≤? Answer? },
{? x？ & gt？ c？ }=? Ω? -{? x？ ≤? c？ }。
So, for any real number? x？ , just know {? x？ ≤? x？ The probability of} is enough, we use? f？ (? x？ ) represents this probability value. Obviously, this probability value is different from? x？ Related, different? x？ , different probability values, the definition of distribution function is given below.
Definition 2? Settings? x？ Is a random variable of any real number. x？ , called function
f？ (? x？ )=? p？ (? x？ ≤? x？ )，-∞& lt； ? x？ & lt+∞
As a random variable? x？ what's up Distribution function? .
For any two real numbers-∞
p？ (? Answer? & lt？ x？ ≤? b？ )=? f？ (? b？ )-? f？ (? Answer? )。
So, as long as it is known? x？ Can you know the distribution function? x？ Fall in any range (? Answer? ,? b？ ], so the distribution function can completely describe the statistical regularity of a random variable.
As can be seen from this definition:
The (1) distribution function is defined on (-∞, +∞), and its value is [0,1];
(2) Any random variable? x？ Everyone has one and only one distribution function. Using the distribution function, the relationship between random variables and random variables can be calculated. x？ Probabilistic problems of related events.
Example 1? Suppose there are 10 balls in a box, of which 5 balls are marked with the number 1, 3 balls are marked with the number 2, and 2 balls are marked with the number 3. Choose a ball from them and record it as a random variable? x？ Expressed as "the number marked on the ball obtained", q? x？ Distribution function of? f？ (? x？ )。
Solution? According to the meaning of the question, random variables? x？ We can take 1, 2,3. According to the calculation formula of classical probability, we can know that the corresponding probability values are 0.5, 0.3 and 0.2 respectively.
What is the definition of distribution function? f？ (? x？ )=? p？ (? x？ ≤? x？ ), therefore
What time? x？ & lt 1, probability? p？ (? x？ ≤? x？ )=0;
When 1≤ x？ & lt2 point, probability? p？ (? x？ ≤? x？ )=? p？ (? x？ = 1)=0.5;
When 2≤? x？ & lt3 points, probability? p？ (? x？ ≤? x？ )=? p？ (? x？ = 1)+? p？ (? x？ =2)=0.5+0.3=0.8;
What time? x？ When ≥3, random events {? x？ ≤? x？ } is an inevitable event, so? p？ (? x？ ≤? x？ ) = 1, i.e.
p？ (? x？ ≤? x？ )=? p？ (? x？ = 1)+? p？ (? x？ =2)+? p？ (? x？ =3)=0.5+0.3+0.2= 1。
Finishing is available? x？ The distribution function of is
f？ (? x？ ) As shown in Figure 2. 1, it is a step curve. x？ There are right continuous jumping points at the three possible values of 1, 2 and 3. What is the height of each jump? x？ The probability of this value point.
Figure 2. 1? f？ (? x？ ) graphics
From the distribution function in the example 1 and its graph, we can see that the distribution function has the properties of right continuity and monotonicity. Specifically, any distribution function? f？ (? x？ ) has the following properties:
(1) For any real number? x？ , there are below 0? f？ (? x？ )≤ 1,?
(2)? f？ (? x？ ) monotonous, that is, when? x？ 1? & lt？ x？ 2? When did you have it? f？ (? x？ 1? )≤? f？ (? x？ 2? );
(3)? f？ (? x？ ) Really? x？ Right continuous function, that is.
Simply prove it.
3. Discrete random variables and their distribution law
Settings? e？ This is a randomized trial. Ω? What is the corresponding sample space? x？ what's up Ω? Random variables open, if? x？ The scope of (recorded as? Ω? x？ ) is it a finite set or a countable set, called? x？ For (one dimension)? Discrete random variable? .
Definition 3? What about one-dimensional discrete random variables? x？ What is the value of? x？ 1? ,? x？ 2? ,…,? x？ n？ , ..., shows the corresponding probability.
p？ (? X=x？ Me? )=? p？ Me? ,? Me? = 1,2,…
Is it a discrete random variable? x？ what's up Distribution law? (or distribution list, probability function).
The distribution law of one-dimensional discrete random variables can also be expressed in the following table.
And satisfy (1) nonnegativity? p？ Me? ≥0,? Me? = 1,2,…; (2) Normative?
? These two properties are also necessary and sufficient conditions for judging whether a sequence can become a distribution law.
Example 2? Set random variables? x？ The distribution law of is as follows.
X- 102
Probability 0.20.40.4
Q ( 1)？ p？ (? x？ ≤-0.7); (2)? x？ Distribution function of? f？ (? x？ )。
Solution? ( 1)? p？ (? x？ ≤-0.7)=? p？ (? x？ =- 1)=0.2。
(2)? x？ Distribution function of? f？ (? x？ ) The solution process is the same as 1, which can be obtained.
From this example, we can know that the distribution function of a discrete random variable can be obtained by knowing its distribution law; On the contrary, if the distribution function of a discrete random variable is known, its distribution law can also be obtained through the following process:
p？ (? x？ =- 1)=? p？ (? x？ ≤- 1)=? f？ (- 1)=0.2,
p？ (? x？ =0)=? p？ (- 1 & lt； ? x？ ≤0)=? f？ (0)-? f？ (- 1)=0.6-0.2=0.4,
p？ (? x？ =2)=? p？ (0 & lt？ x？ ≤2)=? f？ (2)-? f？ (0)= 1-0.6=0.4。
So you can get it? x？ The distribution law of is as follows.
From the above analysis, we can find that the distribution function and the distribution law are equivalent to the description of the value law of discrete random variables. Comparatively speaking, the distribution method is more intuitive and convenient.
Four, continuous random variables and their density functions
The value of continuous random variables fills an interval (or the union of several intervals) on the number axis, and there are infinite uncountable real numbers in this interval. Therefore, when we describe continuous random variables, we can no longer use the distribution law used to describe discrete random variables, but use probability density function instead.
density function
Definition 4? Settings? e？ This is a randomized trial. Ω? What is the corresponding sample space? x？ what's up Ω? Random variable on? f？ (? x？ ) Really? x？ What if there is a non-negative function? f？ (? x？ ) production
What's it called? x？ For (one dimension)? Continuous random variable? ,? f？ (? x？ ) What's your name? x？ what's up (probability) density function? , satisfying: (1) nonnegative f? (? x？ )≥0，-∞& lt； ? x？ & lt+∞; (2) Normative?
Probability density function? f？ (? x？ ) and distribution function? f？ (? x？ ) as shown in figure 2.2,? f？ (? x？ )=? p？ (? x？ ≤? x？ ) it happens to be? f？ (? x？ ) in the interval (-∞,? x？ ], that is, the area of the shaded part in the figure.
Figure 2.2? f？ (? x？ ) What else? f？ (? x？ Geometric relationship of)
Continuous random variables have the following characteristics:
(1) distribution function? f？ (? x？ ) is a continuous function, in? f？ (? x？ ) at a continuous point,? f？ ′(? x？ )=? f？ (? x？ );
(2) For any constant? c？ ，-∞& lt； ? c？ & lt+∞,? p？ (? X=c？ ) =0, so, in the event {? Answer? ≤? x？ ≤? b？ } eliminated? X=a？ Or eliminated? X=b？ , does not affect the size of the probability, that is
p？ (? Answer? ≤? x？ ≤? b？ )=? p？ (? Answer? & lt？ x？ ≤? b？ )=? p？ (? Answer? ≤? x？ & lt？ b？ )=? p？ (? Answer? & lt？ x？ & lt？ b？ )。
It should be noted that this property is invalid for discrete random variables. On the contrary, discrete random variables calculate "point probability".
In addition, this property can also help us to judge whether a non-discrete random variable is a continuous random variable. If a non-discrete random variable has no discrete points and its probability is not 0, it is a continuous random variable.
(3) For any two constants? Answer? ,? b？ ,?
Example 3? Set continuous random variables? x？ The density function of is
Q ( 1)？ p？ (|? x？ | & lt0.5); (2)? x？ Distribution function of? f？ (? x？ )。
Solution? ( 1)?
(2)?
Obviously, it's not hard to find out? f？ (? x？ What is the derivative of? x？ Density function of. f？ (? x？ ) as shown in figure 2.3. It is a continuous curve, and it also satisfies? f？ (? x？ All properties of).
Figure 2.3? f？ (? x？ ) graphics
Exercise 2- 1
1.？ Try to determine the constant? c？ Make the following function a random variable? x？ Distribution law of:
( 1)? p？ (? X=k？ )=? ck？ ,? k？ = 1,…,? n？ ;
(2)?
2.？ Try to determine the constant? c？ , manufacturing?
? Become a random variable x？ The distribution law of, and:
( 1)? p？ (? x？ ≥2);
(2)?
(3)? x？ Distribution function of? f？ (? x？ )。
3.？ There are five balls in a bag, and the five balls are marked with the numbers 1, 2, 3, 4 and 5 respectively. If you don't put any three balls back from this bag, let's assume that the possibility of each ball being taken away is the same, and find the maximum number marked on the ball? x？ Distribution law and distribution function of.
4.？ Known random variables? x？ The distribution law of is as follows. Try to find the unary quadratic equation 3? t？ 2? +2? Xt？ +(? x？ +1)=0 Probability of having real roots.
5.？ Set random variables? x？ The distribution function of is
Beg? x？ Density function and calculate? p？ (? x？ ≤ 1) and? p？ (? x？ & gt2)。
6.？ Known continuous random variable? x？ The distribution function of is
( 1)? Answer? ,? b？ What is the value? f？ (? x？ ) is a continuous function?
(2) Q?
(3) ask? x？ Density function of.
7.？ Set random variables? x？ The density function of is
Find the (1) constant? c？ The value of; (2)?
? (3)? x？ Distribution function of? f？ (? x？ )。
8.？ Set random variables? x？ The density function of is
Find the (1) constant? Answer? The value of; (2)? p？ (- 1 & lt； ? x？ ≤2)。
9.？ Known random variables? x？ The density function of is
Q ( 1)？ p？ (0 & lt？ x？ & lt 1); (2)? x？ The distribution function of.
10.？ Let the lifetime of the transistor (in hours) be a random variable? x？ Its density function is
(1) Try to find the probability that this transistor can't work 150 hours;
(2) An instrument is equipped with four such transistors. Try to find the probability that at least one of the four transistors will fail after the instrument works 150 hours (assuming that the failure of the four transistors does not affect each other).