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1 1

1. 2. 3. 2 2

1 (5/6) 4 =0.517... 5/6 (5/6) 4 1 (5/6) 4 1 (35/36) 24 =0.491... 0.5 2.7 3

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

1 X 1,X 2, m 1 lim n n (X 1 + X 2 +...+ 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 1 +...+X n n 1, 1, 0, 0, 1 3 1 n (X 1 +...+ X n ) p 1 p 1 7

m n m n 50 50 1/4 ( 8

9

[ (Buffon 1707 88) ] p 2/π 2 y 0 y 1 10

1: α 0 α<π π 180 0 α<2π 0 α<π 1 sin(x) 0.8 0.6 α 0.4 y 0.2 0 0 0.5 1 1.5 2 2.5 3 2: sin α sin α y 0 1 0 π 1 π sin α sin α p = 11

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 1 2 4 8 12

2 0 1 2 +21 ( 1 2 )2 +...+2 n ( 1 2 )n +...= 1 2 + 1 2 +... = 1 lim n n (X 1 +...+ X n )= 1 n (X 1 +...+ X n ) log e n. e 2.71... log e 1000 6.91, log e 10000 9.21 13

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

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