ランダムウォークの確率の漸化式と初期条件

Size: px

Start display at page:

Download "ランダムウォークの確率の漸化式と初期条件"

よりとしゆのもと
4 years ago
Views:

1 B L03( Thu) : Time-stamp: Thu 09:16 JST hig X(t), t, t x p(x, t). p(x, t). ( ) L03 B(2019) 1 / 25

2 : L02-Q1 Quiz : 1 X(3) = 1 10 ( ( 3)) = 1., E[X(3)] 1. 2 S 2 = ((3 1)2 + + ( 3 1) 2 ) = E[X(3)] X(3) 3 = 1 10 ( ( 3) 3 ) = E[X(3) 3 ] I [X>1] (X(3)) = ( ) = 10. p = E[I [X>1] (X(3))] L02-Q2 Quiz : ( ) L03 B(2019) 2 / 25

3 : 1 int phi ( int x ){ 2 i f ( x<0){ 3 return 1 ; 4 } 5 return 0 ; 6 } 7 int main ( ) { 8 int n, t, x, count, nmax=1000; / / 9 / scanf, srand / 10 count =0; 11 for ( n=0;n<nmax ; n++){ 12 t =2; x=10; 13 for ( t =3; t <=20; t++){/ 18 / 14 x+=getrandom ( getuniform ( ) ) ; 15 } 16 count+=phi ( x ) ; 17 } 18 p r i n t f ( %f \n, ( double ) count /nmax ) ; 19 return 0 ; 20 } ( ) L03 B(2019) 3 / 25

4 : ( ) L03 B(2019) 4 / 25

5 1 : 2 p(x, t) p(x, t) ( ) L03 B(2019) 5 / 25

6 R(t):. t = 1, 2, 3,... X(t):. X(a) =b ( P (X(a) = b) = 1) X(t) =X(t 1) + R(t) (t = a + 1, a + 2, a + 3,...) R(t) B(1, p) p (r = 1) P (R(t) = r) = q = 1 p (r = 0) 0 ( ) p = 2 3. ( ) L03 B(2019) 6 / 25

7 t , x, t = a, x = b ( 1 ), 2/3 +1, 1/3 X(t) : t ( ) (X(0), X(1), X(2),..., X(t)) : = (path) ( ) x 系列 13 系列 14 系列 15 系列 16 系列 7 系列 8 系列 9 系列 10 系列 11 系列 12 系列 1 系列 2 系列 3 系列 4 系列 5 系列ランダムウォーク, R(t) B(1,2/3) t\x ( ) L03 B(2019) 7 / 25

8 L03-Q1 Quiz( ), X(0) = 0, X(t) = X(t 1) + R(t), (0 < p < 1) p (r = 1) P (R(t) = r) = 1 p (r = 0) 0 ( ) 1 P (X(2) = x) (x = 0, 1,... ). 2 E[X(2)]. 3 V[X(2)]. 4 P (X(2) > 0). cf. ( ) L03 B(2019) 8 / 25

9 ( ) L03 B(2019) 9 / 25

10 t\x t\x ( ) L03 B(2019) 10 / 25

11 p(x, t) 1 : 2 p(x, t) p(x, t) ( ) L03 B(2019) 11 / 25

12 p(x, t) p(x, t) p(x, t) t, x p(x, t) = P (X(t) = x). f(x) = p(x, t) ( t ) t + x= p(x, t) = 1 ( ) L03 B(2019) 12 / 25

13 p(x, t) p(x, t) t x, t + 1, p x + 1, q = 1 p x. X(t) = X(t 1) + R(t) p (r = 1) P (R(t) = r) = q = 1 p (r = 0) 0 ( ) ( 1), x N = t x N p(x, t), t,, x x + 1, N p(x, t 1) p x x N p(x, t 1) q ( ) L03 B(2019) 13 / 25

14 p(x, t) X(t) p(x, t), t, x 1, x, x x,, t x N p(x, t).,!( ),, ( ) L03 B(2019) 14 / 25

15 p(x, t) q q q p p p p x 1 x x t. II(2018)L01 P (X(t) = x) = y P (X(t) = x X(t 1) = y)p (X(t 1) = y) = P (X(t) = x X(t 1) = x 1)P (X(t 1) = x 1) + P (X(t) = x X(t 1) = x)p (X(t 1) = x) =P (R(t) = 1)P (X(t) = x 1) + P (R(t + 1) = 0)P (X(t) = x) ( ) L03 B(2019) 15 / 25

16 p(x, t) 1 : 2 p(x, t) p(x, t) ( ) L03 B(2019) 16 / 25

17 p(x, t) p(x, t) t = 2 x = 3 ( ) L03 B(2019) 17 / 25

18 p(x, t) X(t) p(x, t). 1 p(x, 2) = t = 2 x = 3 X(2) t = 1 x = 0, X(1) p(x, 1) = ( ) L03 B(2019) 18 / 25

19 p(x, t) L03-Q2 Quiz( ) t, x. t = 5 x = 2, t, , 1, 0 ( ). t x p(x, t). ( ) L03 B(2019) 19 / 25

20 p(x, t) L03-Q3 Quiz( ) t, x. t = 3 x = 2, t, x x + 1, x x 2, x. t x p(x, t). ( ) L03 B(2019) 20 / 25

21 1 : 2 p(x, t) p(x, t) ( ) L03 B(2019) 21 / 25

22 p(x, t) I t\x 0 x 1 x x + 1. t 1 p(x 1, t 1) p(x, t 1) p(x + 1, t 1) t p(x 1, t) p(x, t) p(x + 1, t). ( ) L03 B(2019) 22 / 25

23 p(x, t) L03-Q4 Quiz( p(x, t) ). p(x, t) = 2 3 p(x 1, t 1) p(x, t 1), p(x, 0) = { 1 (x = 0) 0 ( ). t\x x t p(x, t) ( ) L03 B(2019) 23 / 25

24 L03-Q5 Quiz( p(x, t) ). { 1 p(x, t) = 5 p(x 1, t 1) + 4 5p(x + 1, t 1) ( ) 0 (x < 1, x > 6), { 0.5 (x = 1, 3) p(x, 0) = 0 ( ). t\x ( ) L03 B(2019) 24 / 25

25 + Learn Math Moodle ryukoku.ac.jp/moodle Moodle App for ios/android URL https: //learn.math.ryukoku.ac. jp/moodle. Math (1-507/1-542) ( ) L03 B(2019) 25 / 25

疎な転置推移確率行列

疎な転置推移確率行列 B E05(2019-05-15 Tue) : Time-stamp: 2019-05-17 Fri 16:18 JST hig http://hig3.net ( ) E05 B(2019) 1 / 11 x = 0,..., m 1 m. p(t), p(x, t) 1 double p [m] = { 1. 0, 0. 0,...., 0. 0 } ; /. m. / 2 / {p ( 0,