32、受限原像随机映射的参数分析

受限原像随机映射的参数分析

1. 映射参数对研究

同时考虑两个映射参数 ξ1 和 ξ2，设 Φ 是 Frn 的随机元素，F(z, u1, u2) 是 r - 映射的生成函数，u1 标记 ξ1，u2 标记 ξ2。相关概率和期望公式如下：
- (P [\xi_1(\Phi) = a_1 \land \xi_2(\Phi) = a_2] = \frac{[z^n u_1^{a_1} u_2^{a_2}] F(z, u_1, u_2)}{[z^n]F(z, 1, 1)})
- (P [\xi_1(\Phi) = a_1|\xi_2(\Phi) = a_2] = \frac{[z^n u_1^{a_1} u_2^{a_2}] F(z, u_1, u_2)}{[z^n u_2^{a_2}]F(z, 1, u_2)})
- (P [\xi_2(\Phi) = a_2|\xi_1(\Phi) = a_1] = \frac{[z^n u_1^{a_1} u_2^{a_2}] F(z, u_1, u_2)}{[z^n u_1^{a_1}]F(z, u_1, 1)})
- (E [\xi_1(\Phi)|\xi_2(\Phi) = a_2] = \frac{[z^n u_2^{a_2}] D_{u_1}F(z, u_1, u_2)| {u_1=1}}{[z^n u_2^{a_2}]F(z, 1, u_2)})
- (E [\xi_2(\Phi)|\xi_1(\Phi) = a_1] = \frac{[z^n u_1^{a_1}] D {u_2}F(z, u_1, u_2)|_{u_2=1}}{[z^n u_1^{a_1}]F(z, u_1, 1)})