4.6-1.ppt - 概率论与数理统计是研究随机现象统 计规律性的学科 随机现象的规律性只有

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概率论与数理统计是研究随机现象统 计规律性的学科 . 随机现象的规律性只有 在相同的条件下进行大量重复试验时才会呈 现出来 . 也就是说,要从随机现象中去寻 求必然的法则,应该研究大量随机现象 .
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研究大量的随机现象,常常采用极限 形式,由此导致对极限定理进行研究 . 限定理的内容很广泛,其中最重要的有两 : 大数定律 中心极限定理 4.6. 大数定律与中心极限定理
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4.6.1 切比雪夫不等式 设随机变量 X 有期望 E ( X ) 和方差 ,则对于 任给 >0, 2 由切比雪夫不等式可以看出,若 越小,则事件 {| X - E ( X )|< } 的概率越大 ,即随机变量 X 集中在期望附近的可能 性越大 . 2 2 2 1 } | ) ( {| X E X P 2 2 } | ) ( {| X E X P 由此可体会方差的概率意义: 它刻划了随机变量取值的离散程度 .
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当方差已知时,切比雪夫不等式给出了 r.v X 与它的期望的偏差不小于 的概率的估 计式 . 如取 3 2 2 } | ) ( {| X E X P 111 . 0 9 } 3 | ) ( {| 2 2 X E X P 可见,对任给的分布,只要期望和方差 存在,则 r.v X 取值偏离 E ( X ) 超过 3 的概率小于 0.111 . 2
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1: 已知正常男性成人血液中,每一毫 升白细胞数平均是 7300 ,均方差是 700 . 利用切比雪夫不等式估计每毫升白细胞数 5200~9400 之间的概率 . 解:设每毫升白细胞数为 X 依题意, E ( X )=7300, D ( X )=700 2 所求为 P (5200 X 9400)
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P(5200 X 9400) = P (5200- 7300 X - 7300 9400- 7300 ) = P (-2100 X - E ( X ) 2100) 2 ) 2100 ( ) ( 1 X D = P { | X - E ( X )| 2100} 由切比雪夫不等式 P { | X - E ( X )| 2100} 2 ) 2100 700 ( 1 9 8 9 1 1
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