概率论与数理统计英文版3精选.pdf

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 X shown is a random variable, it takes values in the set {1,2,6}. Example 3.1.2 The life t of a bulb selected at random from bulbs produced by company A is a random variable, it takes values in the interval (0 , ∞) . 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 27 value or the values in some subset of R. 2. Distribution function Definition 3.1.2 Let X be a random variable on the sample space S . Then the function F (X ) P (X =≤x ) . x ∈R is called the distribution function of X Note The distribution function F (X ) is defined on real numbers, not on sample space. Example 3.1.3 Let X be the number we get from tossing a fair die. Then the distribution function of X is (Figure