离散概率分布
什么是离散随机变量?
- 一个随机变量变量的值是否依赖于函数的结果随机事件
- 随机变量的值在事件执行之前是不知道的(这就是在这种情况下“随机”的含义)
- 随机变量表示为大写字母(
" class="Wirisformula" role="math" alt="X,空格Y”style="vertical-align:-6px;height:22px;width:38px">,等)X , Y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 特定的结果事件的小写字母(
" class="Wirisformula" role="math" alt="X,空格y”style="vertical-align:-6px;height:22px;width:30px">等)x , y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="P左括号X等于X右括号”style="vertical-align:-6px;height:22px;width:64px">表示随机变量的概率P ( X = x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X”style="vertical-align:-6px;height:22px;width:15px">取值X {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x”style="vertical-align:-6px;height:22px;width:11px">”x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 一个离散随机变量(常缩写为DRV)只能取特定的值在一个集合中
- 离散随机变量通常数某物
- 离散随机变量通常只能取有限个值,但也有可能取无限个值(见下面的例子)
- 例子离散随机变量包括:
- 硬币投掷20次后正面朝上的次数
- 它有有限个结果:{0,1,2,…,20}
- 经理在一小时内收到的电子邮件数量
- 这有无限个结果:{1,2,3,…}
- 骰子掷到6点之前的次数
- 这有无限个结果:{1,2,3,…}
- 骰子掷一次时所落的数字
- 它有有限个结果:{1,2,3,4,5,6}
- 硬币投掷20次后正面朝上的次数
离散随机变量的概率分布是什么?
- 一个离散概率分布充分描述所有的值一个离散随机变量可以携带的相关的概率
- 这可以在a中给出表格
- 也可以写成a函数(称为离散概率分布函数或“pdf”)
- 它们可以用垂直线图(横轴上的可能值,纵轴上的概率)
- 的概率之和的所有的值是离散随机变量的1
- 通常是这样写的
" class="Wirisformula" role="math" alt="和直线P左括号X等于X右括号等于1”style="vertical-align:-8px;height:28px;width:112px" loading="lazy">∑ P ( X = x ) = 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 通常是这样写的
- 一个离散均匀分布随机变量取有限个值,每个值都是等概率
- 如果有n个值,那么每个值的概率是
" class="Wirisformula" role="math" alt="1 / n”style="vertical-align:-17px;height:47px;width:19px" loading="lazy">1 n {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 如果有n个值,那么每个值的概率是
如何使用离散概率分布计算概率?
- 第一个画一张表格表示概率分布
- 如果它是一个函数,那么找出每个概率
- 如果任何概率是未知的,那么使用代数来表示它们
- 形成一个方程使用
" class="Wirisformula" role="math" alt="和直线P左括号X等于X右括号等于1”style="vertical-align:-8px;height:28px;width:112px" loading="lazy">∑ P ( X = x ) = 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 把所有的概率加在一起,使总和等于1
- 找到
" class="Wirisformula" role="math" alt="P左括号X = k右括号”style="vertical-align:-6px;height:22px;width:64px" loading="lazy">P ( X = k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="k”style="vertical-align:-6px;height:22px;width:11px" loading="lazy">随机变量的可能值是多少k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X”style="vertical-align:-6px;height:22px;width:15px" loading="lazy">然后X {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="P左括号X = k右括号”style="vertical-align:-6px;height:22px;width:64px" loading="lazy">会在表格中给出吗P ( X = k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="k”style="vertical-align:-6px;height:22px;width:11px" loading="lazy">不是一个可能的值,然后k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="P左括号X = k右括号等于0”style="vertical-align:-6px;height:22px;width:90px" loading="lazy">P ( X = k ) = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 如果
- 找到
" class="Wirisformula" role="math" alt="P左括号X小于等于k右括号”style="vertical-align:-6px;height:22px;width:63px" loading="lazy">P ( X ≤ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 确定所有可能的值,
" class="Wirisformula" role="math" alt="X下标I”style="vertical-align:-12px;height:28px;width:16px" loading="lazy">,这x i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X”style="vertical-align:-6px;height:22px;width:15px" loading="lazy">可以取哪个满足X {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X下标I小于等于k”style="vertical-align:-12px;height:28px;width:42px" loading="lazy">x i ≤ k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 把它们对应的概率加起来
" class="Wirisformula" role="math" alt="直线P左括号X小于等于k右括号等于X下标i小于等于k的直线P左括号X等于X下标i右括号的和”style="vertical-align:-26px;height:46px;width:179px" loading="lazy">P ( X ≤ k ) = ∑ x i ≤ k P ( X = x i ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 一些数学家使用这种符号
" class="Wirisformula" role="math" alt="F左括号x右括号”style="vertical-align:-6px;height:22px;width:33px" loading="lazy">表示累积分布F ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="F左括号x右括号等于P左括号x小于等于x右括号”style="vertical-align:-6px;height:22px;width:112px" loading="lazy">F ( x ) = P ( X ≤ x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 确定所有可能的值,
- 使用类似的方法你可以找到
" class="Wirisformula" role="math" alt="P左括号X小于k右括号”style="vertical-align:-6px;height:22px;width:63px" loading="lazy">,P ( X < k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="P左括号X大于k右括号”style="vertical-align:-6px;height:22px;width:63px" loading="lazy">而且P ( X > k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="P左括号X大于等于k右括号”style="vertical-align:-6px;height:22px;width:63px" loading="lazy">P ( X ≥ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 当所有概率加起来为1时,可以形成以下等价方程:
" class="Wirisformula" role="math" alt="直P左括号X小于k右括号加上直P左括号X等于k右括号加上直P左括号X大于k右括号等于1”style="vertical-align:-6px;height:22px;width:248px" loading="lazy">P ( X < k ) + P ( X = k ) + P ( X > k ) = 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="直线P左括号X大于k右括号等于1减去直线P左括号X小于等于k右括号”style="vertical-align:-6px;height:22px;width:168px" loading="lazy">P ( X > k ) = 1 - P ( X ≤ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="直线P左括号X大于或等于k右括号等于1减去直线P左括号X小于k右括号”style="vertical-align:-6px;height:22px;width:168px" loading="lazy">P ( X ≥ k ) = 1 - P ( X < k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
我怎么知道该用哪个不等式呢?
" class="Wirisformula" role="math" alt="P左括号X小于等于k右括号”style="vertical-align:-6px;height:22px;width:63px" loading="lazy">会被用于这样的短语:P ( X ≤ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 最多不超过,等等
" class="Wirisformula" role="math" alt="P左括号X小于k右括号”style="vertical-align:-6px;height:22px;width:63px" loading="lazy">会被用于这样的短语:P ( X < k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 不到
" class="Wirisformula" role="math" alt="P左括号X大于等于k右括号”style="vertical-align:-6px;height:22px;width:63px" loading="lazy">会被用于这样的短语:P ( X ≥ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 至少,不少于等
" class="Wirisformula" role="math" alt="P左括号X大于k右括号”style="vertical-align:-6px;height:22px;width:63px" loading="lazy">会被用于这样的短语:P ( X > k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 大于等
工作的例子
离散随机变量的概率分布
一)
表明,k = 1 30 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="K = 1 / 30”style="vertical-align:-17px;height:47px;width:54px" loading="lazy">.
b)
计算P ( X ≤ 3 ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="直线P左括号X小于等于3右括号”style="vertical-align:-6px;height:22px;width:62px" loading="lazy">.