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MathML

\begin{aligned} E [Y] & = \exp (μ_{i} + 0.5 ϕ) \frac{Φ (\sqrt{ϕ} - \frac{\log a - μ_{i}}{\sqrt{ϕ}}) - Φ (\sqrt{ϕ} - \frac{\log b - μ_{i}}{\sqrt{ϕ}})}{Φ (\frac{\log b - μ_{i}}{\sqrt{ϕ}}) - Φ (\frac{\log a - μ_{i}}{\sqrt{ϕ}})} \\ V a r [Y] & = \exp (2 μ_{i} + 2 ϕ) \frac{Φ (2 \sqrt{ϕ} - \frac{\log a - μ_{i}}{\sqrt{ϕ}}) - Φ (2 \sqrt{ϕ} - \frac{\log b - μ_{i}}{\sqrt{ϕ}})}{Φ (\frac{\log b - μ_{i}}{\sqrt{ϕ}}) - Φ (\frac{\log a - μ_{i}}{\sqrt{ϕ}})} - {(E [Y])}^{2} \end{aligned}

Math Rendered as SVG

Math as LaTeX Source

\begin{align*} \mr{E}[Y] & = \exp (\mu _ i + 0.5\phi ) \frac{\Phi \left(\sqrt{\phi } - \frac{\log a - \mu _ i}{\sqrt{\phi }} \right) - \Phi \left(\sqrt{\phi } - \frac{\log b - \mu _ i}{\sqrt{\phi }} \right)}{\Phi \left(\frac{\log b - \mu _ i}{\sqrt{\phi }} \right) - \Phi \left(\frac{\log a - \mu _ i}{\sqrt{\phi }} \right)} \\ \mr{Var}[Y] & = \exp (2\mu _ i + 2\phi ) \frac{\Phi \left(2\sqrt{\phi } - \frac{\log a - \mu _ i}{\sqrt{\phi }} \right) - \Phi \left(2\sqrt{\phi } - \frac{\log b - \mu _ i}{\sqrt{\phi }} \right)}{\Phi \left(\frac{\log b - \mu _ i}{\sqrt{\phi }} \right) - \Phi \left(\frac{\log a - \mu _ i}{\sqrt{\phi }} \right)} - \left(\mr{E}[Y] \right)^2 \end{align*}