离散随机变量
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(X, Y,等等)
- 特定的结果事件的小写字母(x, y,等)
" class="Wirisformula" role="math" alt="开始数学大小16px风格直P左括号X等于X右括号结束风格”style="vertical-align:-5px;height:20px;width:58px">表示随机变量的概率X取值P ( X = 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(通常缩写为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
(这有有限数量的结果:0,1,2,…,20) - 经理在一小时内收到的电子邮件数量
(这有无数个结果:1,2,3,…) - 骰子掷到6点之前的次数
(这有无数个结果:1,2,3,…) - 随机抽取的宾果球上的数字
(这有有限数量的结果:1,2,3…,90)
- 硬币投掷20次后正面朝上的次数
概率分布(离散)
什么是概率分布?
- 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(称为概率质量函数)
- 它们可以用垂直线图的可能值X横轴是概率,纵轴是概率)
- 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="开始mathsize 16px样式ΣP左括号X等于X右括号等于1结束样式”style="vertical-align:-5px;height:20px;width:91px" loading="lazy">ΣP ( X = x ) = 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 通常是这样写的
累积概率(离散)
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="开始数学大小16px样式P左括号X = k右括号结束样式”style="vertical-align:-5px;height:20px;width:59px" loading="lazy">P ( X = k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果k随机变量的可能值是多少X然后
" class="Wirisformula" role="math" alt="开始数学大小16px样式P左括号X = k右括号结束样式”style="vertical-align:-5px;height:20px;width:59px" 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="开始mathsize 16px样式k结束样式”style="vertical-align:-4px;height:19px;width:10px" 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="开始数学大小16px样式P左括号X等于k右括号等于0结束样式”style="vertical-align:-5px;height:20px;width:83px" loading="lazy">P ( X = k ) = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 如果k随机变量的可能值是多少X然后
- 找到
" class="Wirisformula" role="math" alt="开始数学大小16px样式P左括号X小于或等于k右括号结束样式”style="vertical-align:-5px;height:20px;width:58px" 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="开始mathsize 16px样式x下标I结束样式”style="vertical-align:-11px;height:26px;width:14px" loading="lazy">,这X可以取哪个满足x i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始数学大小16px样式x下标I小于或等于k结束样式”style="vertical-align:-11px;height:26px;width:38px" 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小于或等于k右括号等于X下标i小于或等于k的直线P左括号X斜体等于X下标i右括号”style="vertical-align:-26px;height:46px;width:178px" loading="lazy">P ( X ≤ k ) = ∑ x i ≤ k P ( X = x i ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 一些数学家使用符号F(x)表示累积分布
" class="Wirisformula" role="math" alt="开始mathsize 16px样式直F左括号x右括号等于直P左括号x小于等于x右括号结束样式”style="vertical-align:-5px;height:20px;width:101px" 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="开始mathsize 16px样式P左括号X小于k右括号逗号空格P左括号X大于等于k右括号空格结束样式”style="vertical-align:-5px;height:20px;width:127px" loading="lazy">和P ( X < k ) , P ( X ≥ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始16px样式P左括号X大于k右括号结束样式”style="vertical-align:-5px;height:20px;width:58px" 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="开始mathsize 16px样式直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结束样式”style="vertical-align:-5px;height:20px;width:222px" loading="lazy">P ( X < k ) + P ( X = k ) + P ( X > k ) = 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:151px" loading="lazy">P ( X > k ) = 1 - P ( X ≤ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:151px" loading="lazy">P ( X ≥ k ) = 1 - P ( X < k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 来计算更复杂的概率,比如
" class="Wirisformula" role="math" alt="开始mathsize 16px风格直P左括号X平方小于4右括号结束风格”style="vertical-align:-5px;height:21px;width:62px" loading="lazy">P ( X 2 < 4 ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 确定括号中随机变量的哪些值满足不等式或事件
- 把相应的概率加起来
我怎么知道该用哪个不等式呢?
" class="Wirisformula" role="math" alt="开始数学大小16px风格直P左括号X小于或等于k右括号结束风格”style="vertical-align:-5px;height:20px;width:57px" loading="lazy">会被用于这样的短语:P ( X ≤ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 最多k,不大于k,等等
" class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:57px" loading="lazy">会被用于这样的短语:P ( X < k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 不到k
" class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:57px" loading="lazy">会被用于这样的短语:P ( X ≥ k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 至少k,不少于k等
" class="Wirisformula" role="math" alt="从MathML转换为可访问文本时出错。”style="vertical-align:-5px;height:20px;width:57px" loading="lazy">会被用于这样的短语:P ( X > k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 大于k等
工作的例子
离散随机变量的概率分布由函数给出
(a)证明这一点k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="k”style="vertical-align:-6px;height:22px;width:11px" loading="lazy">=1 30 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="1 / 30”style="vertical-align:-17px;height:47px;width:27px" loading="lazy">.
(b)计算
(c)
计算P ( X 2 < 5 ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="直线P左括号X方小于5右括号”style="vertical-align:-6px;height:23px;width:69px" loading="lazy">
考试技巧
- 如果离散随机变量可以取的值是有限的,请尝试画一个表
- 在计算概率时,用1减去不想要的值的概率有时比把想要的值的概率加在一起更快
- 始终确保概率在0和1之间,并且它们加起来是1!