From 70f383ec49d7d205e28b97d6968c11987ddf673c Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Mon, 28 Mar 2016 15:49:22 +0200 Subject: [PATCH] Add example for theorem environment --- .../Makefile | 8 ++++ ...rove-transform-random-variable-theorem.pdf | Bin 0 -> 116657 bytes ...rove-transform-random-variable-theorem.tex | 41 ++++++++++++++++++ 3 files changed, 49 insertions(+) create mode 100644 documents/prove-transform-random-variable-theorem/Makefile create mode 100644 documents/prove-transform-random-variable-theorem/prove-transform-random-variable-theorem.pdf create mode 100644 documents/prove-transform-random-variable-theorem/prove-transform-random-variable-theorem.tex diff --git a/documents/prove-transform-random-variable-theorem/Makefile b/documents/prove-transform-random-variable-theorem/Makefile new file mode 100644 index 0000000..1263879 --- /dev/null +++ b/documents/prove-transform-random-variable-theorem/Makefile @@ -0,0 +1,8 @@ +SOURCE=prove-transform-random-variable-theorem + +make: + pdflatex $(SOURCE).tex -output-format=pdf + make clean + +clean: + rm -rf $(TARGET) *.class *.html *.log *.aux *.out diff --git a/documents/prove-transform-random-variable-theorem/prove-transform-random-variable-theorem.pdf b/documents/prove-transform-random-variable-theorem/prove-transform-random-variable-theorem.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a2d082707d9977fe1e10a310aa9bac775e01718d GIT binary patch literal 116657 zcma&NQ>-vduxPn$+qP}n#;5yDGnr)W!|kM!>i6zismdx+1rae? zMmkm~()s0)btq;61_C=nODG;5D0&$aTQg^K0!B_oW`h6wK+%g?SUa0I63~lT8#tSY zm>Ag^n?UjLK{+`)ni$wXxo_TR%GzPGBlLW!Tg#{T(@8sy6K>>-BIfggh-MY?u}C;d z7_9 zO=70WUYL21d~$dAK6G~R__W+eEDO=(-g5P#@^WYt#7t;@Z7<)$iSwg=Ye&N z^Mt0NNw(jGCzB_cFsaJX`(D4Za5Ab1ac*X8ZEUST+%y);yrb0`p`(-gsqA$i;^2Iz zh_R9Y!S1{}-`o}BSd`7mF!-Zl(xqquJKGqMoDr`H-o*J@qhitov4JwSgrGK9@oZb# z6)`z$68ijhj57E6b@(}K{+gP&7+Lvv>B@Ag 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&= \Prob(e^Y \leq t)\\ + &= \begin{cases}\Prob(Y \leq \log(t)) &\text{if } x > 0\\ + 0 &\text{otherwise} + \end{cases} + \end{align} + + Obviously, the density $f_X(x) = 0$ for $x \leq 0$. Now continue with + $t > 0$: + + \begin{align} + \Prob(X \leq t) &= \Prob(Y \leq \log(t))\\ + &= \Phi_{\mu, \sigma^2}(\log(t))\\ + &= \Phi_{0, 1} \left (\frac{\log(t) - \mu}{\sigma} \right)\\ + f_X(x) &= \frac{\partial}{\partial x} \Phi_{0, 1} \left (\frac{\log(x) - \mu}{\sigma} \right)\\ + &= \left (\frac{\partial}{\partial x} \left (\frac{\log(x) - \mu}{\sigma} \right) \right) \cdot \varphi_{0, 1} \left (\frac{\log(x) - \mu}{\sigma} \right)\\ + &= \left (\frac{\sigma \cdot \frac{1}{x}}{\sigma^2} \right) \cdot \varphi_{0, 1} \left (\frac{\log(x) - \mu}{\sigma} \right)\\ + &= \frac{1}{x \sigma} \cdot \varphi_{0, 1} \left (\frac{\log(x) - \mu}{\sigma} \right)\\ + &= \frac{1}{x \sigma} \cdot \frac{1}{\sqrt{2\pi}} \exp \left (-\frac{1}{2} \cdot {\left(\frac{\log(x) - \mu}{\sigma} \right )}^2 \right ) + \end{align} + \end{proof} +\end{document}