\[\begin{align*} E(X) &= \int_0^\infty x \frac{\alpha}{s}\left(\frac{x-m}{s} \right)^{-1-\alpha}\exp\left(-\frac{x-m}{s} \right)^{-\alpha}dx \\ &= \frac{\alpha}{s}\int_0^\infty x \left(\frac{x-m}{s} \right)^{-1-\alpha}\exp\left(-\frac{x-m}{s} \right)^{-\alpha}dx \\ &= \frac{\alpha}{s}\int_0^\infty x \left(\frac{x-m}{s} \right)^{-1}\left(\frac{x-m}{s} \right)^{-\alpha}\exp\left(-\frac{x-m}{s} \right)^{-\alpha}dx \\ \text{let: } u &= \left(\frac{x-m}{s}\right)^{-\alpha} \\ \Rightarrow u^{1/\alpha} &= \left(\frac{x-m}{s}\right)^{-1} \\ \text{and: } x &= \frac{s}{u^{1/\alpha}} + m \\ \therefore dx &= -\frac{s}{\alpha u^{\frac{1}{\alpha}+1}} du \\ E(X) &= \frac{\alpha}{s}\int_\infty^0 (\frac{s}{u^{1/\alpha}} + m) u^{1/\alpha} u e^{-u} \frac{-s}{\alpha u^{\frac{1}{\alpha}+1}} du \\ &= \int_0^\infty (\frac{s}{u^{1/\alpha}} + m) e^{-u} du \\ &= \int_0^\infty \frac{s}{u^{1/\alpha}} e^{-u} du + \int_0^\infty m e^{-u} du \\ &= s\int_0^\infty u^{-1/\alpha} e^{-u} du + m \int_0^\infty e^{-u} du \\ &= s\int_0^\infty u^{(1-1/\alpha)+1} e^{-u} du + m \\ &= m + s\Gamma\left(1-\frac{1}{\alpha}\right) \end{align*}\]
\[\begin{align*} E(X^2) &= \int_0^\infty x^2 \frac{\alpha}{s}\left(\frac{x-m}{s} \right)^{-1-\alpha}\exp\left(-\frac{x-m}{s} \right)^{-\alpha} \\ &= \frac{\alpha}{s}\int_0^\infty x^2 \left(\frac{x-m}{s} \right)^{-1-\alpha}\exp\left(-\frac{x-m}{s} \right)^{-\alpha} \\ &= \frac{\alpha}{s}\int_0^\infty x^2 \left(\frac{x-m}{s} \right)^{-1}\left(\frac{x-m}{s} \right)^{-\alpha}\exp\left(-\frac{x-m}{s} \right)^{-\alpha}dx \\ \text{let: } u &= \left(\frac{x-m}{s}\right)^{-\alpha} \\ \Rightarrow u^{1/\alpha} &= \left(\frac{x-m}{s}\right)^{-1} \\ \text{and: } x &= \frac{s}{u^{1/\alpha}} + m \\ x^2 &= \left(\frac{s}{u^{1/\alpha}} + m\right)^2 \\ &= \left(\frac{s^2}{u^{2/\alpha}} + \frac{2ms}{u^{1/\alpha}}+ m^2\right)\\ \therefore dx &= -\frac{s}{\alpha u^{\frac{1}{\alpha}+1}} du \\ E(X^2) &= \frac{\alpha}{s}\int_\infty^0 \left(\frac{s^2}{u^{2/\alpha}} + \frac{2ms}{u^{1/\alpha}}+ m^2\right) u^{\frac{1}{\alpha}} u e^{-u} \frac{-s}{\alpha u^{\frac{1}{\alpha}+1}} du\\ &= \int_0^\infty \left(\frac{s^2}{u^{2/\alpha}} + \frac{2ms}{u^{1/\alpha}}+ m^2\right) u^{\frac{1}{\alpha}+1-1-\frac{1}{\alpha}} e^{-u} du \\ &= \int_0^\infty \frac{s^2}{u^{2/\alpha}} e^{-u} du + \int_0^\infty \frac{2ms}{u^{1/\alpha}}e^{-u} du + \int_0^\infty m^2e^{-u} du\\ &= s^2\int_0^\infty u^{(1-2/\alpha)-1} e^{-u} du + 2ms \int_0^\infty u^{(1-1/\alpha)-1}e^{-u} du + m^2\\ &= s^2\Gamma\left(1-\frac{2}{\alpha}\right) + 2ms\Gamma\left(1-\frac{1}{\alpha}\right) +m^2 \\ \mathrm{Var}(X) &= s^2\Gamma\left(1-\frac{2}{\alpha}\right) + 2ms\Gamma\left(1-\frac{1}{\alpha}\right) +m^2 - \left[m + s\Gamma\left(1-\frac{1}{\alpha}\right) \right]^2 \\ &= s^2\Gamma\left(1-\frac{2}{\alpha}\right) + 2ms\Gamma\left(1-\frac{1}{\alpha}\right) +m^2 - m^2 -2ms\Gamma\left(1-\frac{1}{\alpha}\right) - s^2\Gamma\left(1-\frac{1}{\alpha}\right)^2 \\ &= s^2[\Gamma(1-\frac{2}{\alpha}) - [\Gamma(1-\frac{1}{\alpha}]^2] \end{align*}\]