added 2 proofs

Author: JoramSoch

P/cf-fct.md

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+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-09-22 09:12:00
+
+title: "Characteristic function of a function of a random variable"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Probability functions"
+theorem: "Characteristic function of arbitrary function"
+
+sources:
+  - authors: "Taboga, Marco"
+    year: 2017
+    title: "Functions of random vectors and their distribution"
+    in: "Lectures on probability and mathematical statistics"
+    pages: "retrieved on 2021-09-22"
+    url: "https://www.statlect.com/fundamentals-of-probability/functions-of-random-vectors"
+
+proof_id: "P260"
+shortcut: "cf-fct"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ be a [random variable](/D/rvar) with the [expected value](/D/mean) function $\mathrm{E}_X[\cdot]$. Then, the [characteristic function](/D/cf) of $Y = g(X)$ is equal to
+
+$$ \label{eq:cf-fct}
+\varphi_Y(t) = \mathrm{E}_X \left[ \mathrm{exp}(it \, g(X)) \right] \; .
+$$
+
+
+**Proof:** The [characteristic function](/D/cf) is defined as
+
+$$ \label{eq:cf}
+\varphi_Y(t) = \mathrm{E} \left[ \mathrm{exp}(it \, Y) \right] \; .
+$$
+
+Due of the [law of the unconscious statistician](/P/mean-lotus)
+
+$$ \label{eq:mean-lotus}
+\begin{split}
+\mathrm{E}[g(X)] &= \sum_{x \in \mathcal{X}} g(x) f_X(x) \\
+\mathrm{E}[g(X)] &= \int_{\mathcal{X}} g(x) f_X(x) \, \mathrm{d}x \; ,
+\end{split}
+$$
+
+$Y = g(X)$ can simply be substituted into \eqref{eq:cf} to give
+
+$$ \label{eq:cf-fct-qed}
+\varphi_Y(t) = \mathrm{E}_X \left[ \mathrm{exp}(it \, g(X)) \right] \; .
+$$

P/mgf-fct.md

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+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-09-22 09:00:00
+
+title: "Moment-generating function of a function of a random variable"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Probability functions"
+theorem: "Moment-generating function of arbitrary function"
+
+sources:
+  - authors: "Taboga, Marco"
+    year: 2017
+    title: "Functions of random vectors and their distribution"
+    in: "Lectures on probability and mathematical statistics"
+    pages: "retrieved on 2021-09-22"
+    url: "https://www.statlect.com/fundamentals-of-probability/functions-of-random-vectors"
+
+proof_id: "P259"
+shortcut: "mgf-fct"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ be a [random variable](/D/rvar) with the [expected value](/D/mean) function $\mathrm{E}_X[\cdot]$. Then, the [moment-generating function](/D/mgf) of $Y = g(X)$ is equal to
+
+$$ \label{eq:mgf-fct}
+M_Y(t) = \mathrm{E}_X \left[ \mathrm{exp}(t \, g(X)) \right] \; .
+$$
+
+
+**Proof:** The [moment-generating function](/D/mgf) is defined as
+
+$$ \label{eq:mgf}
+M_Y(t) = \mathrm{E} \left[ \mathrm{exp}(t \, Y) \right] \; .
+$$
+
+Due of the [law of the unconscious statistician](/P/mean-lotus)
+
+$$ \label{eq:mean-lotus}
+\begin{split}
+\mathrm{E}[g(X)] &= \sum_{x \in \mathcal{X}} g(x) f_X(x) \\
+\mathrm{E}[g(X)] &= \int_{\mathcal{X}} g(x) f_X(x) \, \mathrm{d}x \; ,
+\end{split}
+$$
+
+$Y = g(X)$ can simply be substituted into \eqref{eq:mgf} to give
+
+$$ \label{eq:mgf-fct-qed}
+M_Y(t) = \mathrm{E}_X \left[ \mathrm{exp}(t \, g(X)) \right] \; .
+$$