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概率論與數(shù)理統(tǒng)計(jì)中的常用公式(一)

2023-07-13 14:24 作者:格心致力  | 我要投稿

1、加法公式:對(duì)于任意兩事件A,B有

P%EF%BC%88A%5Ccup%20B%EF%BC%89%3DP(A)%2BP(B)-P(AB)

2、乘法公式:設(shè)P(A)%3E0,則有

P(AB)%3DP(B%7CA)P(A)

3、全概率公式:設(shè)試驗(yàn)E的樣本空間為S,AE的事件,B_1,B_2,...,B_nS的一個(gè)劃分,且P(B_i)%3E0(i%3D1%2C2%2C...%2Cn),則

P%5Cleft(A%5Cright)%3D%5Csum_%7Bi%3D1%7D%5E%7Bn%7DP%5Cleft(A%7CB_i%5Cright)P%5Cleft(B_i%5Cright)

4、貝葉斯公式:設(shè)試驗(yàn)E的樣本空間為S,AE的事件,B_1,B_2,...,B_nS的一個(gè)劃分,且P(A)%3E0P(B_i)%3E0(i%3D1%2C2%2C...%2Cn),則

P%5Cleft(B_i%7CA%5Cright)%3D%5Cfrac%7BP%5Cleft(A%7CB_i%5Cright)P%5Cleft(B_i%5Cright)%7D%7B%5Csum_%7Bj%3D1%7D%5E%7Bn%7DP%5Cleft(A%7CB_j%5Cright)P%5Cleft(B_j%5Cright)%7Di%3D1%2C2%2C...%2Cn

5、高斯積分

%5Cint_%7B%EF%B9%A3%E2%88%9E%7D%5E%7B%E2%88%9E%7Dexp(-%5Cfrac%7Bt%5E2%7D%7B2%7D)%3D%5Csqrt%7B2%5Cpi%7D

(口訣:負(fù)二分方作指數(shù),超越數(shù)e作底數(shù);兩者合體作被積,從窮到窮2π根)

6、卷積公式

%20f_x*f_y%3D%5Cint_%7B%EF%B9%A3%E2%88%9E%7D%5E%7B%E2%88%9E%7D%20f_y%5Cleft(z-x%5Cright)f_x%5Cleft(x%5Cright)dx%3D%5Cint_%7B%EF%B9%A3%E2%88%9E%7D%5E%7B%E2%88%9E%7D%20f_x%5Cleft(z-y%5Cright)f_y%5Cleft(y%5Cright)dy

7、期望均值

⑴離散型期望:加權(quán)平均

⑵連續(xù)型期望:積分平均

E%5Cleft(x%5Cright)%3D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20xf(x)dx

8、方差

⑴定義式

D%5Cleft(x%5Cright)%3DE%5Cleft(%5Cleft(x-E%5Cleft(x%5Cright)%5Cright)%5E2%5Cright)

⑵離散型方差:加權(quán)方差

⑶連續(xù)型方差:積分方差

D%5Cleft(x%5Cright)%3D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20%5Cleft(x-E%5Cleft(x%5Cright)%5Cright)%5E2f%5Cleft(x%5Cright)dx

(均值協(xié)方差的特殊形式)

9、協(xié)方差

?⑴定義式

Cov%5Cleft(x%2Cy%5Cright)%3DE%5Cleft%5C%7B%5Cleft%5Bx-E%5Cleft(x%5Cright)%5Cright%5D%5Cleft%5By-E%5Cleft(y%5Cright)%5Cright%5D%5Cright%5C%7D%0A%0A

⑵和方差公式:

D%5Cleft(x%2By%5Cright)%3DD%5Cleft(x%5Cright)%2B2Cov%5Cleft(x%2Cy%5Cright)%2BD%5Cleft(y%5Cright)

(口訣:首方差,尾方差,二倍協(xié)差在中央)

⑶均值協(xié)方差:

Cov%5Cleft(x%2Cy%5Cright)%3DE%5Cleft(xy%5Cright)-E%5Cleft(x%5Cright)E%5Cleft(y%5Cright)

(口訣:協(xié)方差,積均值,去均積)


概率論與數(shù)理統(tǒng)計(jì)中的常用公式(一)的評(píng)論 (共 條)

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