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\begin{aligned} μ_{d i f f} & = δ σ (N w_{1} w_{2})^{- \frac{1}{2}} + μ_{0} \\ μ_{1} & = δ σ (N w_{1} w_{2})^{- \frac{1}{2}} + μ_{0} - μ_{2} \\ μ_{2} & = δ σ (N w_{1} w_{2})^{- \frac{1}{2}} + μ_{0} - μ_{1} \\ σ & = {\begin{cases} δ^{- 1} (N w_{1} w_{2})^{\frac{1}{2}} (μ_{d i f f} - μ_{0}), & | δ | > 0 \\ undefined, & otherwise \end{cases} \\ w_{1} & = {\begin{cases} \frac{1}{2} \pm \frac{1}{2} {[1 - \frac{4 δ^{2} σ^{2}}{N (μ_{d i f f} - μ_{0})^{2}}]}^{\frac{1}{2}}, & 0 < | δ | \leq \frac{1}{2} N^{\frac{1}{2}} \frac{| μ_{d i f f} - μ_{0} |}{σ} \\ undefined, & otherwise \end{cases} \\ w_{2} & = {\begin{cases} \frac{1}{2} \pm \frac{1}{2} {[1 - \frac{4 δ^{2} σ^{2}}{N (μ_{d i f f} - μ_{0})^{2}}]}^{\frac{1}{2}}, & 0 < | δ | \leq \frac{1}{2} N^{\frac{1}{2}} \frac{| μ_{d i f f} - μ_{0} |}{σ} \\ undefined, & otherwise \end{cases} \end{aligned}

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\begin{align*} \mu _\mr {diff} & = \delta \sigma (N w_1 w_2)^{-\frac{1}{2}} + \mu _0 \\ \mu _1 & = \delta \sigma (N w_1 w_2)^{-\frac{1}{2}} + \mu _0 - \mu _2 \\ \mu _2 & = \delta \sigma (N w_1 w_2)^{-\frac{1}{2}} + \mu _0 - \mu _1 \\ \sigma & = \left\{ \begin{array}{ll} \delta ^{-1} (N w_1 w_2)^\frac {1}{2} (\mu _\mr {diff} - \mu _0), & |\delta | > 0 \\ \mbox{undefined}, & \mbox{otherwise} \\ \end{array} \right. \\ w_1 & = \left\{ \begin{array}{ll} \frac{1}{2} \pm \frac{1}{2} \left[ 1 - \frac{4 \delta ^2 \sigma ^2}{N(\mu _\mr {diff} - \mu _0)^2} \right]^\frac {1}{2}, & 0 < |\delta | \le \frac{1}{2}N^\frac {1}{2} \frac{|\mu _\mr {diff} - \mu _0|}{\sigma } \\ \mbox{undefined}, & \mbox{otherwise} \\ \end{array} \right. \\ w_2 & = \left\{ \begin{array}{ll} \frac{1}{2} \pm \frac{1}{2} \left[ 1 - \frac{4 \delta ^2 \sigma ^2}{N(\mu _\mr {diff} - \mu _0)^2} \right]^\frac {1}{2}, & 0 < |\delta | \le \frac{1}{2}N^\frac {1}{2} \frac{|\mu _\mr {diff} - \mu _0|}{\sigma } \\ \mbox{undefined}, & \mbox{otherwise} \\ \end{array} \right. \\ \end{align*}