机器算法验证 - 计算随机游走的稳态概率向量{ 0 , 1 , ... , n }{0,1,…,n} - 吾爱随笔录

假设我们在上进行随机游走，具有转移概率 $\{0, \dots, n\}$

P (x, x + 1) = p P (x, x - 1) = 1 - p

$P(x, x + 1) = p \\ P(x, x - 1) = 1 - p$

对于 , , ,和 $1 \le x \le n -1$ $P(0, 1) = a$ $P(0, 0) = 1 - a$ $P(n, n - 1) = b$ $P(n, n) = 1- b$

我需要计算这个随机游走的稳态向量。首先我写下五个差分方程 $\pi$

π (x + 1) (1 - p) + π (x - 1) p = π (x) 2 \leq x \leq n - 2

$\pi(x + 1)(1-p) + \pi(x - 1)p = \pi(x) \ \ \ \ \ \ \ 2 \le x \le n -2$

π (0) (1 - a) + π (1) (1 - p) = π (0)

$\pi(0)(1 - a) + \pi(1)(1 - p) = \pi(0)$

π (0) a + π (2) (1 - p) = π (1)

$\pi(0)a + \pi(2)(1 - p) = \pi(1)$

π (n - 1) p + π (n) (1 - b) = π (n)

$\pi(n-1)p + \pi(n)(1 - b) = \pi(n)$

π (n - 2) p + π (n) b = π (n - 1)

$\pi(n - 2)p + \pi(n)b = \pi(n-1)$

我们也有要求

\sum_{k = 0}^{n} π (x) = 1

$\sum_{k=0}^n \pi(x) = 1$

第一个方程的通解是

π (x) = k_{1} + k_{2} (\frac{p}{p - 1})^{x}, p \neq 1 / 2

$\pi(x) = k_1 + k_2\bigg(\frac{p}{p-1}\bigg)^x , \ \ \, p \ne 1/2$

π (x) = k_{1} + k_{2} x, p = 1 / 2

$\pi(x) = k_1 + k_2x, \ \ \ \ p = 1/2$

我不确定下一步该怎么做。任何帮助表示赞赏。