Xの期待値は，
\begin{align}
\mathbb{E}_{X\sim f_X}[X]&= \int_{-\infty}^\infty\exp\{-\frac{|x-\mu|}{\sigma}\}dx\\
&= \int_{-\infty}^\mu\frac{x}{2\sigma}\exp(\frac{x-\mu}{\sigma}dx+\int_{\mu}^\infty \frac{x}{2\sigma}\exp(-\frac{x-\mu}{\sigma})dx\\
&= \int_{-\infty}^\mu\frac{x}{2}\{\exp(\frac{x-\mu}{\sigma})\}'dx+\int_\mu^\infty\frac{x}{2}\{-\exp(\frac{x-\mu}{\sigma})\}'dx\\
&= \left[\frac{x}{2}\exp(\frac{x-\mu}{\sigma})\right]_{-\infty}^\mu-\int_{-\infty}^\mu\frac{1}{2}\exp(\frac{x-\mu}{\sigma})dx\\
& \ \ \ +\left[-\frac{x}{2}\exp(-\frac{x-\mu}{\sigma})\right]_\mu^\infty+\int_\mu^\infty\frac{1}{2}\exp(-\frac{x-\mu}{\sigma})dx\\
&= \mu-\frac{\sigma}{2}+\frac{\sigma}{2}\\
&= \mu
\end{align}
分散は，
\begin{align}
Var(X)&= \int_{-\infty}^\infty\frac{(x-\mu)^2}{2\sigma}\exp\{-\frac{|x-\mu|}{\sigma}\}dx\\
&= \int_{-\infty}^\mu\frac{(x-\mu)^2}{2\sigma}\exp(\frac{x-\mu}{\sigma})dx\\
&\ \ \ +\int_\mu^\infty\frac{(x-\mu)^2}{2\sigma}\exp\{-\frac{(x-\mu)}{\sigma}\}dx\\
&= \int_{-\infty}^\mu\frac{(x-\mu)^2}{2}\{\exp(\frac{x-\mu}{\sigma})\}'dx\\
&\ \ \ -\int_\mu^\infty\frac{(x-\mu)^2}{2}\{\exp(\frac{-x+\mu}{\sigma})\}'dx\\
&= \left[\frac{(x-\mu)^2}{2}\exp(\frac{x-\mu}{\sigma})\right]_{-\infty}^\mu-\int_{-\infty}^\mu(x-\mu)\exp(\frac{x-\mu}{\sigma})dx\\
&\ \ \ -\left[\frac{(x-\mu)^2}{2}\exp(\frac{-x+\mu}{\sigma})\right]_\mu^\infty+\int_\mu^\infty(x-\mu)\exp(\frac{-x+\mu}{\sigma})dx\\
&= -\int_{-\infty}^\mu(x-\mu)\exp(\frac{x-\mu}{\sigma})dx+\int_\mu^\infty(x-\mu)\exp(\frac{-x+\mu}{\sigma})dx\\
&= -\int_{-\infty}^\mu\sigma(x-\mu)\{\exp(\frac{x-\mu}{\sigma})\}'dx-\int_\mu^\infty\sigma(x-\mu)\{\exp(\frac{-x+\mu}{\sigma})\}'dx\\
&= -\left[\sigma(x-\mu)\exp(\frac{x-\mu}{\sigma})\right]_{-\infty}^\mu+\int_{-\infty}^\mu\sigma\exp(\frac{x-\mu}{\sigma})dx\\
&\ \ \ -\left[\sigma(x-\mu)\exp(\frac{-x+\mu}{\sigma})\right]_\mu^\infty+\int_\mu^\infty\sigma\exp(\frac{-x+\mu}{\sigma})dx\\
&= \left[\sigma^2\exp(\frac{x-\mu}{\sigma})\right]_{-\infty}^\mu+\left[\sigma^2\exp(\frac{-x+
\mu}{\sigma})\right]_\mu^\infty\\
&= 2\sigma^2
\end{align}
参考文献
- 『現代数理統計学の基礎』(久保川達也 著)