概率论与数理统计(英文).docVIP

3. Random Variables 3.1 Definition of Random Variables In 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 certain company, or the weight of cows in a certain farm, etc. These ideas lead to the definition of random variables. 1. random variable definition Definition 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 space. 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 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 function Definition 3.1.2 Let be a random variable on the sample 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 function in Example 3.1.3 3. Properties The 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 3.1.4 Let be the life of au