4.3.ppt - 4.3 几个重要随机变量的 数学期望及方差 1 0-1 分布 设随机变量 X 具有 0 1 分布 其分布律为 P X 0 1 p P X 1 p E X

4.3.ppt - 4.3 几个重要随机变量的...

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4.3 几个重要随机变量的 数学期望及方差 1. 0-1 分布 设随机变量 X 具有 1 0 分布 , 其分布律为 , 1 } 0 { p X P , } 1 { p X P , 1 ) 1 ( 0 ) ( p p p X E , 1 ) 1 ( 0 ) ( 2 2 2 p p p X E 2 2 )] ( [ ) ( ) ( X E X E X D ). 1 ( 2 p p p p
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2. 二项分布 X ~ B ( n , p ), X 表示 n 重贝努里试验中 “ 成功” 次数 . 若设 次试验失败 如第 次试验成功 如第 i i X i 0 1 i =1,2 ,…, n n 次试验中“成功” 的次数 n i i X X 1 = np 因为 P ( X i =1)= p , P ( X i =0)= 1- p n i i X E 1 ) ( 所以 E ( X )= E ( X i )= ) 1 ( 0 1 p p = p
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D ( X i )= E ( X i 2 )-[ E ( X i )] 2 E ( X i )= P ( X i =1)= p , E ( X i 2 )= p , = p - p 2 = p (1- p ) 于是 i =1,2 ,…, n 由于 X 1 , X 2 ,…, X n 相互独立 n i i X D X D 1 ) ( ) ( = np (1- p ) 可见,服从参数为 n p 的二项分布 的随机变量 X 的数学期望是 np , 方差是 np (1 - p ).
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3. 泊松分布 ), ( ~ P X X 的分布律为 , ! } { k e k X P k , 0 , , 2 , 1 , 0 k 0 ! ) ( k k k e k X E 0 1 )! 1 ( k k k e , e e ] ) 1 ( [ ) ( 2 X X X E X E ) ( )] 1 ( [ X E X X E 0 !
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