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1、 3. Random Variables3.1 Definition of Random VariablesIn engineering or scientific problems, we are not only interested in the probability of events, but also interested in some variables depending on sample points. （定义在样本点上的变量）For example, we maybe interested in the life of bulbs produced by a cert
2、ain company, or the weight of cows in a certain farm, etc. These ideas lead to the definition of random variables.1. random variable definitionDefinition 3.1.1 A random variable is a real valued function defined on a sample space; i.e. it assigns a real number to each sample point in the sample spac
3、e.Here are some examples.Example 3.1.1 A fair die is tossed. The number shown is a random variable, it takes values in the set . Example 3.1.2 The life of a bulb selected at random from bulbs produced by company A is a random variable, it takes values in the interval . Since the outcomes of a random
4、 experiment can not be predicted in advance, the exact value of a random variable can not be predicted before the experiment, we can only discuss the probability that it takes some value or the values in some subset of R.2. Distribution functionDefinition 3.1.2 Let be a random variable on the sample
5、 space . Then the function . is called the distribution function of Note The distribution function is defined on real numbers, not on sample space.Example 3.1.3 Let be the number we get from tossing a fair die. Then the distribution function of is (Figure 3.1.1) Figure 3.1.1 The distribution functio
6、n in Example 3.1.33. PropertiesThe distribution function of a random variable has the following properties:(1) is non-decreasing.In fact, if , then the event is a subset of the event ,thus (2), .(3)For any , .This is to say, the distribution function of a random variable is right continuous.Example
7、3.1.4 Let be the life of automotive parts produced by company A , assume the distribution function of is (in hours)Find ,.Solution By definition, . Question： What are the probabilities and ?Example 3.1.5 A player tosses two fair dice, if the total number shown is 6 or more, the player wins $1, other
8、wise loses $1. Let be the amount won, find the distribution function of .Solution Let be the total number shown, then the events contains sample points, . Thus , And so Thus Figure 3.1.2 The distribution function in Example 3.1.5The distribution function of random variables is a connection betweenpr
9、obability and calculus. By means of distribution function, the main tools in calculus, such as series, integrals are used to solve probability and statistics problems.3.2 Discrete Random Variables 离散型随机变量In this book, we study two kinds of random variables.Definition 3.2.1 A random variable is calle
10、d a discrete random variable, if it takes values from a finite set or, a set whose elements can be written as a sequence Assume a discrete random variable takes values from the set . Let , (3.2.1)Then we have , . the probability distribution of the discrete random variable (概率分布) X a1 a2 anprobabili
11、ty p1 p2 pn注意随机变量X的分布所满足的条件(1) Pi 0(2) P1+P2+Pn=1离散型分布函数And the distribution function of is given by (3.2.2)In general, it is more convenient to use (3.2.1) instead of (3.2.2). Equation (3.2.1) is called the probability distribution of the discrete random variable .Example 1For an experiment in whic
12、h a coin is tossed three times (or 3 coins are tossed once), construct the distribution of X. (Let X denote the number of head occurrence)Solution n=3, p=1/2X pr 0 1/81 3/82 3/83 1/8Example 2在一次试验中，事件A发生的概率为p, 不发生的概率为1p, 用X=0表示事件A没有发生，X=1表示事件A发生，求X的分布。two-point distribution(两点分布) X01P1-pp某学生参加考试得5分的
13、概率是p, X表示他首次得5分的考试次数，求X的分布。geometric distribution （几何分布） X 1234kPpq1pq2pq3pqk1pExample 3 (射击5发子弹) 某射手有5发子弹，射一次命中率为0.9,如果命中目标就停止射击，如果不命中则一直射到子弹用尽，求耗用子弹数x的概率分布。*Example 3.2.1 A die is tossed, by we denote the number shown, Assume that the probability is proportional to , . Find the probability distrib
14、ution of .Solution Assume that , constant, .Since the events , are mutually exclusive and their union is the certain event, i.e., the sample space , we have ,thus . The probability distribution of is (Figure 3.2.1) , . Figure 3.2.1 Probability distribution in Example 3.2.1Question. What is the difference between distribution functions and probability distributions例2 有一种验血新方法：
