27 / 25 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 3.1.1A fair die is tossed. The number Xshown is a random variable, it takes values in the set {1,2,6}L. WExample 3.1.2The 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 ,) . WSince 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 28 / 25 value or the values in some subset of R. 2. Distribution function Definition 3.1.2Let Xbe a random variable on the sample space S. Then the function ()()F XP Xx . Rxis called the distribution function of XNote The distribution function ()F Xis defined on real numbers, not on sample space.Example 3.1.3Let Xbe the number we get from tossing a fair die. Then the distribution function of Xis (Figure 3.1.1) 0,1;( ),1,1,2,,5;61,6.if xnF xif nxnnif xLFigure 3.1.1 The distribution function in Example 3.1.3 3. PropertiesThe distribution function ( )F xof a random variable Xhas th...
