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1 1 1

2

3 1 (5/6) 4 = /6 (5/6) 4 1 (5/6) 4 1 (35/36) 24 =

4 1 n =rand() 0 1 = rand() () rand 6 0,1,2,3,4, *6 int() integer 1 6 = int(rand()*6) % 51.7% 3 [ ] 3/4 4

5 ( )/6 = = 5000 ( ) 5, =50 5

6 2 X a 1,...,a n p 1,...,p n a 1 p a n p n (mean) (expectation) X E[X] E(X) [ ] 0 1 4,1 1 2,2 1 4, E[X] = =1 4 X =0, 1, 1, 2 E[X] = 1 { } =1 4 6

7 1 X 1,X 2, m 1 lim n n (X 1 + X X n )=m p E[X] =0 P (X =0)+1 P (X =1)=0 (1 p)+1 p = p E[X] p X 1,X 2,... X X n n 1, 1, 0, 0, n (X X n ) p 1 p 1 7

8 m n m n /4 ( 8

9 9

10 [ (Buffon ) ] p 2/π 2 y 0 y 1 10

11 1: α 0 α<π π α<2π 0 α<π 1 sin(x) α 0.4 y : sin α sin α y π 1 π sin α sin α p = 11

12 π p = 1 π π 0 sin xdx = 1 π [ ] π cos x = 2 0 π N n N n/n p =2/π 2(N/n) π π 1 1 π/4 N n n/n π/4 π 4n/N =rand() 4 n 2 n

13 ( 1 2 ) n ( 1 2 )n +...= = 1 lim n n (X X n )= 1 n (X X n ) log e n. e log e , log e

14 5 ( ) 1 Y 1,Y 2,... Y k k k Y 1,Y 2,...,Y k ((k +1) ) Y k+1 E[Y k+1 ] {Y 1,Y 2,...,Y k } E[Y k+1 Y 1,...,Y k ] E[Y k+1 Y 1,...,Y k ] Y k k Y k = E[Y k+1 Y 1,...,Y k ], k =1, 2,... Y k E[Y k+1 Y 1,...,Y k ], k =1, 2,... 14

15 [ ] [ ] [ ] [ ] ( ) 15

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