truncated Normal moments

An interesting if presumably hopeless question spotted on X validated: a lower-truncated Normal distribution is parameterised by its location, scale, and truncation values, μ, σ, and α. There exist formulas to derive the mean and variance of the resulting distribution,  that is, when α=0,

\Bbb{E}_{\mu,\sigma}[X]= \mu + \frac{\varphi(\mu/\sigma)}{1-\Phi(-\mu/\sigma)}\sigma


\text{var}_{\mu,\sigma}(X)=\sigma^2\left[1-\frac{\mu\varphi(\mu/\sigma)/\sigma}{1-\Phi(-\mu/\sigma)}  -\left(\frac{\varphi(\mu/\sigma)}{1-\Phi(-\mu/\sigma)}\right)^2\right]

but there is no easy way to choose (μ, σ) from these two quantities. Beyond numerical resolution of both equations. One of the issues is that ( μ, σ) is not a location-scale parameter for the truncated Normal distribution when α is fixed.

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