\[\begin{align*} E(X) &= \int_{-\infty}^\infty x\frac{1}{\beta}\exp\left(-\frac{x-\mu}{\beta}-\exp\left(-\frac{x-\mu}{\beta}\right)\right)dx \\ &= \frac{1}{\beta}\int_{-\infty}^\infty \exp\left(-\frac{x-\mu}{\beta}-\exp\left(-\frac{x-\mu}{\beta}\right)\right)dx \\ \text{let: } z&= \frac{x-\mu}{\beta} \\ \Rightarrow dz &= \frac{dx}{\beta} \\ \therefore dx &= \beta dz \\ \text{and: } x &= \mu +\beta z \\ E(X) &= \int_{-\infty}^\infty (\mu +\beta z)e^{-z - e^{-z}}dz \\ &= \mu\int_{-\infty}^\infty e^{-z - e^{-z}}dz + \beta\int_{-\infty}^\infty z e^{-z - e^{-z}}dz \\ &= \mu\int_{-\infty}^\infty e^{-z}e^{-e^{-z}}dz + \beta\int_{-\infty}^\infty z e^{-z}e^{-e^{-z}}dz \\ \text{let: } e^{-z} &= u \\ du &= -e^{-z}dz \\ \Rightarrow dz &= -\frac{du}{u} \\ \text{and: } z &= -\ln u \\ E(X) &= \mu\int_{\infty}^0 ue^{-u}\frac{-du}{u} + \beta\int_0^\infty -\ln u \cdot u e^{-u}\frac{-du}{u} \\ &= \mu\int_0^{\infty} e^{-u}du + \beta\int_0^\infty \ln u e^{-u}du \\ &= \mu + \beta\gamma \end{align*}\]
Note: Euler-Mascheroni Constant:
\[\begin{align*} \gamma &= \lim_{n\to\infty} \left(\sum_{k=1}^n \frac{1}{k} -\log n\right) \\ &= \int_1^\infty \left(\frac{1}{\lfloor x \rfloor} - \frac{1}{x} \right)dx \\ &= 0.57721 \dots \end{align*}\]
\[\begin{align*} E(X^2) &= \int_{-\infty}^\infty x^2\frac{1}{\beta}\exp\left(-\frac{x-\mu}{\beta}-\exp\left(-\frac{x-\mu}{\beta}\right)\right)dx \\ &= \frac{1}{\beta}\int_{-\infty}^\infty \exp\left(-\frac{x-\mu}{\beta}-\exp\left(-\frac{x-\mu}{\beta}\right)\right)dx \\ \text{let: } z&= \frac{x-\mu}{\beta} \\ \Rightarrow dz &= \frac{dx}{\beta} \\ \therefore dx &= \beta dz \\ \text{and: } x &= \mu +\beta z \\ \Rightarrow x^2 &= (\mu^2 + 2\mu\beta z + \beta^2 z^2) \\ E(X^2) &= \frac{1}{\beta}\int_{-\infty}^\infty (\mu^2 + 2\mu\beta z + \beta^2 z^2) e^{-z - e^{-z}} \beta dz \\ &= \mu^2 \int_{-\infty}^\infty e^{-z - e^{-z}} dz + 2\mu\beta \int_{-\infty}^\infty z e^{-z - e^{-z}} dz + \beta^2\int_{-\infty}^\infty z^2 e^{-z - e^{-z}} dz\\ \text{let: } u &= e^{-z} \\ \Rightarrow dz &= -\frac{du}{u} \\ \text{and: } z &= -\ln u \\ \Rightarrow z^2 &= \ln^2 u \\ E(X^2) &= \mu^2 \int_\infty^0 ue^{-u}\frac{-du}{u} + 2\mu\beta \int_0^\infty -\ln u ue^{-u} \frac{-du}{u} + \beta^2\int_\infty^0 (\ln^2 u) ue^{-u} \frac{-du}{u}\\ &= \mu^2 \int_\infty^0 e^{-u}du + 2\mu\beta \int_0^\infty (\ln u) e^{-u}du + \beta^2\int_0^\infty (\ln^2 u) e^{-u} du\\ &= \mu^2 + 2\mu\beta\gamma + \beta^2\gamma^2 + \beta^2\frac{\pi^2}{6} \\ \mathrm{Var}(X) &= \mu^2 + 2\mu\beta\gamma + \beta^2\gamma^2 + \beta^2\frac{\pi^2}{6} - \mu^2 - 2\mu\beta\gamma - \beta^2\gamma^2 \\ &= \beta^2\frac{\pi^2}{6} \end{align*}\]
\[\begin{align*} M_X(t) &= \int_{-\infty}^\infty e^{tx} \frac{1}{\beta} e^{-\left(\left(\frac{x-\mu}{\beta}\right) + e^{\frac{x-\mu}{\beta}} \right)} dx \\ \text{let: } z &= \frac{x-\mu}{\beta} \\ x &= \mu + z\beta \\ \Rightarrow dx &= \beta dz \\ M_X(t) &= \int_{-\infty}^\infty e^{t(\mu + z\beta)} e^{-z-e^{-z}} dz \\ &= \int_{-\infty}^\infty e^{\mu t} e^{z(\beta t -1)} e^{-e^{-z}} dz \\ \text{let: } u &= e^{-z} \\ \Rightarrow dz &= \frac{-du}{u} \\ \text{and: } z &= -\ln u \\ M_X(t) &= e^{\mu t} \int_{\infty}^0 e^{(\beta t -1)(-\ln u)} e^{-u} \frac{-du}{u} \\ &= e^{\mu t} \int_0^{\infty} u^{(1-\beta t)-1} e^{-u} du \\ &= e^{\mu t} \int_0^{\infty} u^{(-\beta t)} e^{-u} du \\ &= e^{\mu t} \Gamma(1-\beta t) \end{align*}\]