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(类似于GCSE)
- 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{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X"style="vertical-align:-6px;height:22px;width:15px" loading="lazy">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ΣP(X=x)=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="ΣP左括号X = X右括号= 1"style="vertical-align:-5px;height:20px;width:91px" loading="lazy">
- 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1n{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="1 / n"style="vertical-align:-14px;height:37px;width:15px" loading="lazy">
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∑P(X=x)=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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">
- 找到P(X=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 = k右括号结束样式"style="vertical-align:-5px;height:20px;width:59px" loading="lazy">
- If是随机变量的一个可能值X然后P(X=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 = 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
- 如果k{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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P(X=k)=0{"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">
- 找到P(X≤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小于或等于k右括号结束样式"style="vertical-align:-5px;height:20px;width:58px" loading="lazy">
- 确定所有可能的值,xi{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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xi≤k{"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">
- 把它们对应的概率加起来
- P(X≤k)=∑xi≤kP(X=xi){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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">
- 一些数学家使用符号F(x)表示累积分布
- F(x)=P(X≤x){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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">
- 使用类似的方法你可以找到P(X<k), P(X≥k) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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">而且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">
- 当所有概率加起来为1时,可以形成以下等价方程:
- P(X<k)+P(X=k)+P(X>k)=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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)=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"}" class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:151px" loading="lazy">
- 来计算更复杂的概率,比如P(X2<4){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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">
- 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?
- P(X≤k){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" 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"}" 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"}" 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"}" class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:57px" loading="lazy">会被用于这样的短语:
离散随机变量的概率分布由函数给出
P(X=x)=kx2 x=-3,-1,2,40 otherwise.{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="直P左括号X等于X右括号等于开花括号表属性columnalign左端属性行单元格k X的平方空格空格空格空格空格X等于- 3,- 1,2,4结束单元格行单元格0空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格空格否则。结束单元格结束表关闭"style="vertical-align:-21px;height:55px;width:260px" loading="lazy">
(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">=
130{"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:67px" loading="lazy">
(c)
计算
P(X2<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">
- 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!