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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

16 bold playing 16

, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x

, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x 1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d

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, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f ,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)

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1 1 1 7 1.1.................................. 11 2 13 2.1............................ 13 2.2............................ 17 2.3.................................. 19 3 21 3.1.............................

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