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我们已经介绍了随机变量的数学期望,它体现了随机变量取值的平均水平,是随机变量的一个重要的数字特征 . PowerPoint PPT Presentation


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我们已经介绍了随机变量的数学期望,它体现了随机变量取值的平均水平,是随机变量的一个重要的数字特征. 但是在一些场合,仅仅知道平均值是不够的. 甲仪器测量结果. 较好. 乙仪器测量结果. 例如,某零件的真实长度为 a ,现用甲、乙两台仪器各测量 10 次,将测量结果 X 用坐标上的点表示如图:. 测量结果的均值都是 a. 若让你就上述结果评价一下两台仪器的优劣,你认为哪台仪器好一些呢?. 因为乙仪器的测量结果集中在均值附近. 中心. 中心. 乙炮. 又如 , 甲、乙两门炮同时向一目标射击 10 发炮弹,其落点距目标的位置如图:. 甲炮射击结果. 乙炮射击结果.

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我们已经介绍了随机变量的数学期望,它体现了随机变量取值的平均水平,是随机变量的一个重要的数字特征 .

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6002134

.

.


6002134

a10X

a


6002134

,10

?

.


6002134

,.


6002134

XE[(X-E(X)]2<

D(X)=E[X-E(X)]2 (1)

X.

X-E(X)

X.


6002134

D(X)=E[X-E(X)]2

.

X

X .

D(X)=0,r.v X1 .


6002134

X

g(X)=[X-E(X)]2 .

X

P(X=xk)=pk

X

X~f(x)


6002134

D(X)=E(X2)-[E(X)]2

D(X)=E[X-E(X)]2

=E{X2-2XE(X)+[E(X)]2}

=E(X2)-2[E(X)]2+[E(X)]2

=E(X2)-[E(X)]2

.


6002134

1 r.vX

P(X=k)=p(1-p)k-1, k=1,2,n

0<p<1,D(X)

q=1-p


6002134

+E(X)

D(X)=E(X2)-[E(X)]2


6002134

X1 X2

D(X1 +X2)=

1. C,D(C)=0;

2. C,D(CX)=C2D(X);

3. X1X2

D(X1+X2)= D(X1)+D(X2);

X1,X2,,Xn,


6002134

4.D(X)=0 P(X= C)=1 C=E(X)

P(X= x)

.


6002134

i=1,2n

n

2

X~B(n,p), Xn

.

E(Xi)=P(Xi=1)= p,

E(Xi2)= p,

D(Xi)= E(Xi2)-[E(Xi)]2

= p- p2= p(1- p)


6002134

D(Xi)= E(Xi2)-[E(Xi)]2 = p- p2= p(1- p)

i=1,2n

X1,X2,,Xn

= np(1- p)


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XE(X)

>0,

{|X-E(X)|< }X.

.


6002134


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r.vX .

r.v XE(X) 3 0.111 .


6002134

P(5200 X 9400)

3 7300700 . 5200~9400 .

X

E(X)=7300,D(X)=7002


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P(5200 X 9400)

=P(5200-7300 X-7300 9400-7300)

= P(-2100 X-E(X) 2100)

= P{ |X-E(X)| 2100}

P{ |X-E(X)| 2100}

5200~94008/9 .


6002134

4 A 0.75, nn, A0.74~0.760.90?

XnA

X~B(n, 0.75)

E(X)=0.75n,

D(X)=0.75*0.25n=0.1875n

n .


6002134

n

P(0.74n< X<0.76n )

=P(-0.01n<X-0.75n< 0.01n)

= P{ |X-E(X)| <0.01n}

= P{ |X-E(X)| <0.01n}


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n 18750n

, A0.74~0.76

0.90 .


6002134

.

.

r.v


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