m*******s
发帖数: 758 | 1 10个红球和8个蓝球随机排成一行，
what is the expected number of the balls which has a neighbor with different
color? |
m*******s
发帖数: 758 | 2 靠，全在考虑c++了，这个可是正儿八经的数学考试第一题……
哎哎,高手呢？
different
【在 m*******s 的大作中提到】
: 10个红球和8个蓝球随机排成一行，
: what is the expected number of the balls which has a neighbor with different
: color?
|
m*******s
发帖数: 758 | 3 B(m,n)= mn(3m+3n-2)/(m+n-1)/(m+n);
different
【在 m*******s 的大作中提到】
: 10个红球和8个蓝球随机排成一行，
: what is the expected number of the balls which has a neighbor with different
: color?
|
d******e
发帖数: 152 | 4 Please list in detail
【在 m*******s 的大作中提到】
: B(m,n)= mn(3m+3n-2)/(m+n-1)/(m+n);
:
: different
|
K*****Y
发帖数: 629 | 5 Both computer simulation & analytical derivation gives 13.5948 instead. |
m*p
发帖数: 6 | 6 This is the correct answer
【在 K*****Y 的大作中提到】
: Both computer simulation & analytical derivation gives 13.5948 instead.
|
k***e
发帖数: 556 | 7 I get 13.54. Hope it is correct.
By using the linearty of expection, we can calculate the exp of all
positions. Just notice that the first and last positions have the same exp
and is different from the 16 positions in the middle
Both computer simulation & analytical derivation gives 13.5948 instead.
【在 K*****Y 的大作中提到】
: Both computer simulation & analytical derivation gives 13.5948 instead.
|
e**********n
发帖数: 359 | 8 Let x_k = +/- 1, +1 for red, -1 for blue, suppose we have n red balls and m
blue balls.
Define
T_1 = (1-x_1 x_2)/2
...
T_i = (3-x_{i-1} x_{i+1} - x_{i-1}x_i - x_i x_{i+1})/4
...
T_{n+m} = (1- x_{n+m}x_{n+m-1})/2.
T_k = 0,1, 1 for 'having neighboring ball(s) of different color', 0 for 'no
neighboring balls of different color'.
N = \sum_{k=1}^{n+m} T_k.
E(N) = (3n+3m - 2)/4 ( 1- E(x_i x_j))
Due to permutation symmetry, the index i and j are arbitrary as long as i!=j.
The problem is reduced to ca |
k***e
发帖数: 556 | 9 Wow, delicate construction of T_i.
would you mind tell me how you get this great idea?
m
no
【在 e**********n 的大作中提到】
: Let x_k = +/- 1, +1 for red, -1 for blue, suppose we have n red balls and m
: blue balls.
: Define
: T_1 = (1-x_1 x_2)/2
: ...
: T_i = (3-x_{i-1} x_{i+1} - x_{i-1}x_i - x_i x_{i+1})/4
: ...
: T_{n+m} = (1- x_{n+m}x_{n+m-1})/2.
: T_k = 0,1, 1 for 'having neighboring ball(s) of different color', 0 for 'no
: neighboring balls of different color'.
|
e**********n
发帖数: 359 | 10 This is similar to Ising model with nearest neighbor and next nearest
neighbor interactions.
【在 k***e 的大作中提到】
: Wow, delicate construction of T_i.
: would you mind tell me how you get this great idea?
:
: m
: no
|
