7.1 理解分辨随机变量

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  理解异异随机变量

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  What is the purpose of a random variable? How do random variables differ from algebraic variables?
  ::随机变量的目的是什么? 随机变量与代数变量有何不同?

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  Look to the end of the lesson for the answer.
  ::寻找教训的结尾 以找到答案。

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  Understanding Discrete Random Variables
  ::理解异异随机变量

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  You are familiar with variables through your extensive use of them in Algebra. In this lesson we are going to introduce the concept of   random variables . A random variable assigns a unique   numerical value to the outcome of a random experiment . Note how I pointed out the importance that the values are numerical and that they are chosen randomly. It is also important to note that the total probability of all of the possible values of the random variable should be 100%.
  ::您通过在代数中广泛使用变量而熟悉变量。 在这个课程中, 我们将引入随机变量的概念。 一个随机变量会为随机实验的结果指定一个独特的数字值。 注意我如何指出数值是数字的重要性, 并且它们是随机选择的。 同样重要的是要注意随机变量所有可能值的总概率应该是100% 。

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  Random variables are often used to represent the number of times you get a specific result of a random process. For instance, the random variable  $\begin{align*}C\end{align*}$ might represent the number of times you get a total of nine by rolling two six-sided dice three times.
  ::随机变量通常用来表示随机过程得出特定结果的次数。例如,随机变量C可能代表通过滚动两张六面骰子三次获得总共9次的次数。

  

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  A  discrete random variable is a random variable with a specific and countable set of possible values. If a set is defined as a series of specific values, whether there are 5, or 10, or 100,000,000 values, that set could be described by a discrete random variable.
  ::离散随机变量是一个随机变量,其中含有一套具体和可计算可能的值。如果一组被定义为一系列特定值,无论是5值还是10值,还是10 000 000值,或者10 000 000值,那么该集合可以用一个离散随机变量来描述。

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  Identifying Random Variables 
  ::识别随机变量

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  1. If   $\begin{align*}T\end{align*}$  is the outcome of flipping a fair coin once, is   $\begin{align*}T\end{align*}$  a random variable?
  ::1. 如果T是一次抛出一个公平硬币的结果,T是随机变数吗?

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  No,  $\begin{align*}T\end{align*}$ is not a random variable, because it is not a   numeric result. The outcome of flipping a coin would be either “heads” or “tails”. To make $\begin{align*}T\end{align*}$ a random variable, you would have to set a number for each outcome, such as:   $\begin{align*}T=1\end{align*}$ if heads and 2 if tails.
  ::否, T不是一个随机变量, 因为它不是一个数字结果。 翻硬币的结果要么是“ 头 ” , 要么是“ 尾巴 ” 。 要将 T 变成随机变量, 您就必须为每个结果设定一个数字, 如: T=1, 如果头和尾为2, 则为 T=1 。

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  2. If Trina designates   $\begin{align*}Y\end{align*}$  to be the number of yellow marbles she gets during nine trials of randomly pulling 1 marble from a bag filled with marbles of various colors and returning it, is   $\begin{align*}Y\end{align*}$ a random variable?
  ::2. 如果Trina指定Y是她在9次随机抽取一个大理石试验中得到的黄色大理石的数量,从装满各种颜色的大理石的袋子中抽取一个大理石,然后还回去,Y是随机变数吗?

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  Yes,   $\begin{align*}Y\end{align*}$  is a random variable, since it is the random numerical result of a limited number of independent trials of an experiment.
  ::是的,Y是一个随机变量,因为它是对实验进行数量有限的独立试验的随机数字结果。

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  Finding Unknown Values 
  ::查找未知值

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  If   $\begin{align*}N\end{align*}$  is the number of nines you get when rolling two standard dice three times:
  ::如果 N 是滚动双标准骰子三次时得到的 9 个数 :

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  a. Is  $\begin{align*}N\end{align*}$  a binomial random variable?
  ::a. N是一个二进制随机变数吗?

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  $\begin{align*}N\end{align*}$  is a binomial random variable, because it is the result of a specific and limited number of independent trials of a random process and each outcome is either nine or not nine.
  ::N是一个二进制随机变量,因为它是随机过程的个别和有限的独立试验的结果,每个结果有9个或9个。

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  b. What are the possible values of  $\begin{align*}N\end{align*}$ ?
  ::b. N的可能值是什么?

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  Since you could only roll a total of 9 once each trial ,   $\begin{align*}N\end{align*}$  could be 0, 1, 2, or 3.
  ::因为每次审判,你只能卷9次, N可能是0, 1, 2, 或3次。

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  c. What is the of  $\begin{align*}N\end{align*}$ ?
  :c) 什么是N?

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  Since there are only four ways that a 9 could be rolled:   $\begin{align*}6 + 3\end{align*}$ ,   $\begin{align*}5 + 4\end{align*}$ ,   $\begin{align*}4 + 5\end{align*}$ , and   $\begin{align*}3 + 6\end{align*}$ , out of the  $\begin{align*}6 \times 6 = 36\end{align*}$   possible combinations, your chances of getting a 9 on any given roll would be $\begin{align*}\frac{4}{36} = \frac{1}{9}\end{align*}$ . The probabilities of each of the possible values of  $\begin{align*}N\end{align*}$ would be:
  
  ::由于一个9的滚动方式只有四种:6+3、5+4、4+5和3+6,在6x6=36可能的组合中,你获得一个9的机会为436=19。

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  • $\begin{align*}N=0\end{align*}$  (You do not roll 9 at all):   $\begin{align*}\frac{8}{9} \times \frac{8}{9} \times \frac{8}{9} =\frac{512}{729}\end{align*}$
    ::N=0(你根本不滚9):89x89x89=5127
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  • $\begin{align*}N=1\end{align*}$  (You roll a single 9, no more or less):   $\begin{align*}\frac{1}{9} \times \frac{8}{9} \times \frac{8}{9}=\frac{64}{729} \times 3=\frac{192}{729}\end{align*}$
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    • Multiply by 3, because there are 3 possible positions for the single “9”: 1 ^st , 2 ^nd , or 3 ^rd
      ::乘以3乘以3,因为单一“9”有3个可能的位置:第一、第二或第三
    
    ::N=1(您滚动一个9,不增或减):19x89x89=64729x3=192729乘以3,因为单“9”有3个可能的位置:第一、第二或第三
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  • $\begin{align*}N=2\end{align*}$  (You roll 9 exactly twice):   $\begin{align*}\frac{1}{9} \times \frac{1}{9} \times \frac{8}{9}=\frac{8}{729} \times 3=\frac{24}{729}\end{align*}$
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    • Multiply by 3, because there are 3 possible positions for the single “not 9”
      ::乘以3乘以3,因为单一“非9”有3个可能的职位
    
    ::N=2( 您滚9 的滚9 精确两次): 19x19x89= 8729x3= 24729 乘以 3 乘以 3, 因为单“ 不是 9” 有 3 个可能的位置 。
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  • $\begin{align*}N=3\end{align*}$ (Three 9’s in a row): $\begin{align*}\frac{1}{9} \times \frac{1}{9} \times \frac{1}{9}=\frac{1}{729}\end{align*}$
    ::N=3(连续三九):19x19x19x19=1729

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  Earlier  Problem Revisited
  ::重审先前的问题

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  What is the purpose of a random variable? How do random variables differ from algebraic variables?
  ::随机变量的目的是什么? 随机变量与代数变量有何不同?

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  A random variable is used to simplify the expression of the possible outcomes of a random process. Random variables differ from algebraic variables most prominently in that algebraic variables commonly only represent a single value, where as random variables represent a range of possible numeric outcomes   and the associated probability of each.
  ::随机变量用于简化随机过程可能结果的表达方式。随机变量与代数变量的差异最大,因为代数变量通常只代表单一值,随机变量代表各种可能的数值结果和相关概率。

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  Examples 
  ::实例

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  Example 1
  ::例1

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  If random variable  $\begin{align*}A\end{align*}$  represents the age of a single randomly chosen student from your classroom, is  $\begin{align*}A\end{align*}$  a discrete random variable?
  ::如果随机变数 A 代表您教室中单人随机选择的学生的年龄, 是一个离散随机变数吗 ?

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  No,   $\begin{align*}A\end{align*}$  is not discrete because age could represent any of an infinite number of values based on how accurately you measure.
  ::否, A 不是独立的, 因为年龄可以代表一个无限数量的值, 根据您如何精确的测量值 。

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  Example 2
  ::例2

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  If  $\begin{align*}B\end{align*}$  is the age of a randomly chosen student in your classroom, is  $\begin{align*}B\end{align*}$  a discrete random variable?
  ::如果B是课堂上随机选择的学生的年龄,B是一个离散随机变量吗?

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  No,   $\begin{align*}B\end{align*}$  is not discrete because although there must be a limited number of students in the classroom, the number of possible ages is not countable (15 years, 5 months, 3 days, 2 hours, 10 minutes and 1 second… or 2 seconds… or 2.33 seconds, etc.)
  ::否,B不是分开的,因为尽管课堂上的学生人数必须有限,但可能的年龄数无法计算(15年5个月3天、2小时、10分钟1秒、2秒、2秒、2.33秒等)。

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  Example 3
  ::例3

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  If  $\begin{align*}C\end{align*}$  is defined by the function  $\begin{align*}C=\frac{102}{3}\end{align*}$ , is  $\begin{align*}C=\frac{102}{3}\end{align*}$  a discrete random variable?
  ::如果C由函数 C=1023 定义, C=1023 是离散随机变量吗 ?

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  No, based on the given function,   $\begin{align*}C\end{align*}$  is not random since it will always represent 34.
  ::否,根据给定的函数,C不是随机的,因为它总是代表34。

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  Review 
  ::回顾

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  For questions 1-10, state why the example does or does not describe a random variable:
  ::对于问题1-10,请说明为什么该例子没有或没有说明随机变量:

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  1. $\begin{align*}K\end{align*}$  is the number of Kings you get over 10 trials of randomly drawing 1 card at a time from a standard deck.
     ::K是标准甲板上每次随机抽取1张牌的10次以上测试的国王人数。
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  2. $\begin{align*}S\end{align*}$  is the number of cards you draw before drawing the Seven of Hearts.
     ::S 是绘制红心七之前您绘制的牌数。
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  3. One person is chosen at random from a classroom,   $\begin{align*}H\end{align*}$  is her height in inches.
     ::从教室随机挑选一个人,H是她的身高,英寸。
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  4. A fair coin is flipped 10 times,   $\begin{align*}T\end{align*}$  is the number of tails.
     ::一个公平的硬币被翻了10倍,T是尾巴的数量。
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  5. $\begin{align*}B\end{align*}$  is the number of barks you hear in the city park during one-sixth of an hour (10 minutes), the sixth of an hour you choose to listen is selected by roll of a fair die.
     ::B 是指在城市公园听到的每小时六分之六(10分钟)的树皮数量, 选择聆听的每小时的第六小时, 是通过一个公平死亡的滚动选择的。
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  6. $\begin{align*}S\end{align*}$  is the sound you hear first during a ten minute period at the park, the 10 minute period is chosen by the roll of a fair die.
     ::S是您在公园10分钟时间里第一次听到的声音, 10分钟时间由公平死亡的卷子所选择。
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  7. $\begin{align*}N\end{align*}$  is the number that makes the statement   $\begin{align*}N+4=9 \end{align*}$  true.
     ::N 是使声明N+4=9成为真实的编号。
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  8. Choose a number between 1 and 10, multiply it by the age, in years, of the person next to you and add your own age in years. Do this for 5 people.  $\begin{align*}A\end{align*}$ is the average of the results from the six trials.
     ::选择 1 和 10 之间的数字, 乘以您旁边的人的年龄, 以年数乘以您旁边的人的年龄, 加上您自己的年龄 。 5 人使用这个数字。 A 是六个试验结果的平均结果 。
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  9. $\begin{align*}H\end{align*}$  is the average maximum vertical jump of the girls on the girls’ basketball team.
     ::H是女孩篮球队中女孩的平均最高垂直跳跃。
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  10. $\begin{align*}H\end{align*}$  is the number of jumps greater than 22 inches in a basketball game.
      ::H是篮球比赛中超过22英寸的跳跃次数。

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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 。