7.2 了解连续随机变量

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  理解连续随机变量

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  If  $\begin{array}{r}H\end{array}$ is the height of 5 boys chosen at random from your school. Is  $\begin{array}{r}H\end{array}$ a continuous random variable or is it discrete? What is it about the experiment that makes $\begin{array}{r}H\end{array}$ continuous or discrete? What could you do to change it?
  ::如果 H 是 5 个男孩的身高, 是 从 学校 随机选取的 。 哈 是 连续 随机 变数 还是 离 散 ? 实验 使 H 连续 或 离 散 的 是 什么呢 ? 您能做些什么来改变 ?

  

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  Understanding Continuous Random Variables 
  ::理解连续随机变量

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  Recall that  random variables assign numeric values to the outcomes of independent random events. A discrete random variable is used to represent a specific and countable number of values, such as the number of people in a room, or the number of times you roll a four during ten rolls of a fair die .
  ::回顾随机变量为独立随机事件的结果指定数字值。一个离散随机变量用来代表特定和可计算的数量的值,例如一个房间的人数,或者在公平死亡的十卷中你滚动四次的次数。

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  A  continuous random variable is used to represent all of the possible values in a particular interval, such as the distance around a randomly chosen lake, or the weight of a randomly chosen rock from a pile. In either of these cases, the result could literally be an infinite number of possibilities, since the exact distance around the lake (around each molecule of water), and the exact weight of the rock (to the smallest of measures) is impossible to pinpoint.
  ::连续随机变量用于代表特定间隔内的所有可能值,如随机选择的湖泊周围的距离,或从堆积中随机选择的岩石的重量。 在这两种情况下,结果都可能是无限的可能性,因为湖周围的准确距离(围绕每个水分子)和岩石的准确重量(到最小的量度)无法确定。

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  In short: If the possible values of a random variable are countable, it is a discrete random variable. If the values are uncountable, it is a continuous random variable.
  ::简言之:如果随机变量的可能值可以计算,则该变量是一个离散随机变量。如果数值无法计算,则是一个连续随机变量。

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  Identifying Continuous Random Variables
  ::识别连续随机变量

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  If Brian uses the variable  $\begin{array}{r}Y\end{array}$ to represent the top bicycling speed of a randomly chosen student in class, is  $\begin{array}{r}Y\end{array}$ a continuous random variable?
  ::如果Brian使用变数Y代表班里随机选择的学生的顶级自行车速度,Y是一个连续随机变量吗?

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  Yes, speed is a common continuous variable, and the value is chosen by a random process. We know it is continuous because there is always another possible value between any two speed values. It would not be possible to count all of the possible speeds that  $\begin{array}{r}Y\end{array}$ could be.
  ::是的, 速度是一个共同的连续变量, 值是由随机过程选择的。 我们知道它是连续的, 因为任何两个速度值之间总有另一个可能的值。 无法计算 Y 可能的所有速度 。

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  Understanding Continuous Random Variables 
  ::理解连续随机变量

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  1. Would the arm length of a randomly chosen preschooler be represented by a continuous random variable?
  ::1. 随机选择的学龄前儿童的平长是否由连续随机变量代表?

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  Yes, length is another common continuous variable. There is always another possible value for length between any two values.
  ::是的, 长度是另一个常见的连续变量。 任何两个值之间的长度总是有另一个可能的值 。

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  2. Would a continuous variable be used to represent the number of people in your city?
  ::2. 是否使用一个连续变量来代表贵国城市的人数?

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   No, even though there may be a huge number, the number of people in your city is finite, and there are no fractions or decimals of people.
  ::没有,尽管可能数量很大,但你们城市中的人数是有限的,没有几分数或十进制。

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  Earlier Problem Revisited
  ::重审先前的问题

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  If   $\begin{array}{r}H\end{array}$ is the total height of 5 boys chosen at random from your school. Is  $\begin{array}{r}H\end{array}$ a continuous random variable or is it discrete? What is it about the experiment that makes  $\begin{array}{r}H\end{array}$ continuous or discrete? What could you do to change it?
  ::如果 H 是 5 个男孩的完全身高 。 H 是 连续 随机 变量 还是 分离 ? 它与 使 H 连续 或 离 异 的 实验有什么关系? 您能做些什么来改变它 ?

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  $\begin{array}{r}H\end{array}$  is a continuous random variable. We know it is continuous because the possible height values aren’t countable. One way to make  $\begin{array}{r}H\end{array}$ a discrete random variable would be to say that  $\begin{array}{r}H=1\end{array}$ if the total heights were less than 25 feet,  $\begin{array}{r}H=2\end{array}$ if between 25 and 30 feet, and  $\begin{array}{r}H=3\end{array}$ if more than 30 feet. Then  $\begin{array}{r}H\end{array}$ would be discrete with the possible values 1, 2, and 3, which are quite countable.
  ::H 是一个连续随机变量。 我们知道它是连续的, 因为可能的高度值是不可计数的。 将 H 变成一个离散随机变量的一个方法就是说, 如果总高度小于25英尺, H=1; 如果在25至30英尺之间, H=2; 如果在30英尺以上, H=3。 然后H 与可能的1、 2和3值分离, 这些值是相当可计数的。

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  Examples 
  ::实例

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  Example 1
  ::例1

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  If  $\begin{array}{r}B\end{array}$ represents the mean time spent baking a batch of cookies by 5 students chosen at random from Home Ec class, is  $\begin{array}{r}B\end{array}$ a continuous random variable?
  ::如果B代表5名随机选自家教班的学生做一批饼干的平均时间, B是一个连续随机变数吗?

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  Yes, time is continuous, and the result of the experiment is a random value.
  ::是的,时间是连续的,实验的结果是一个随机值。

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  Example 2
  ::例2

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  Would you use a continuous or discrete random variable to represent the number of atoms in a baseball?
  ::您是否使用连续或离散随机变量来表示棒球中的原子数量?

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  Even though there is a seemingly uncountable number of atoms, the number is finite. Therefore, this would be a discrete variable.
  ::尽管原子的数量似乎难以计算,但数量是有限的。因此,这将是一个独立的变量。

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  Example 3
  ::例3

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  Roll a fair die three times, the total is the number of minutes you run water from a hose into a small pool. Would the volume of water in the pool be represented by a continuous or discrete random variable? What about the number of minutes you run the hose?
  ::滚动会死三次, 总计是您将水从水管流到小水池的分钟数。 水池中的水量会由连续或离散随机变量代表吗? 您运行水管的分钟数如何 ?

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  Volume is continuous, so the amount of water would be represented by a continuous random variable. The number of minutes is countable, so it would be a discrete variable.
  ::水量是连续的, 所以水量会用一个连续随机变量来表示。 分钟数是可计数的, 因此它会是一个独立的变量 。

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  Example 4
  ::例4

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  Roll two dice, the total is the number of people you count before asking the next how many games they purchased. If  $\begin{array}{r}C\end{array}$ is the number of dollars he or she spent, is  $\begin{array}{r}C\end{array}$ discrete or continuous?
  ::滚动两骰子, 总数是您在询问接下来的游戏购买多少次之前计算的人数。 如果 C 是他或她花费的美元数, C 是离散的还是连续的 ?

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  $\begin{array}{r}C\end{array}$  is discrete, money is measured in specific denominations 
  ::C是分立的,货币以特定单位计量。

  

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  Review 
  ::回顾

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  For questions 1-10, assume the “student” in the situation is chosen at random, and state whether the variable is discrete or continuous.
  ::对于问题1-10,假设情况中的“学生”是随机选择的,并说明变量是离散的还是连续的。

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  1. The body temperature of the student.
     ::学生的体温
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  2. The number of brothers or sisters the student has.
     ::学生的兄弟姐妹人数。
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  3. The exact age of the student.
     ::学生的确切年龄
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  4. The number of classes the student signed up for this semester.
     ::学生报名参加本学期的班级数量。
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  5. The number of books the student has in his or her bag.
     ::学生在包里藏的书籍数量。
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  6. The student’s favorite teacher.
     ::学生最喜爱的老师。
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  7. The weight of the student.
     ::学生的体重
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  8. The percentage of students in the school taller than the student, to the nearest 1%.
     ::在校学生中比学生高的百分比,接近1%。
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  9. The student’s G.P.A.
     ::学生G.P.A.
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  10. The mean number of people in the student’s classes.
      ::学生班的平均人数。

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。