addd proof "mgf-sumind"

Author: JoramSoch

P/mgf-sumind.md

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+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-11-08 10:46:00
+
+title: "Moment-generating function of a sum of independent random variables"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Other probability functions"
+theorem: "Moment-generating function of sum of independents"
+
+sources:
+  - authors: "Probability Fact"
+    year: 2021
+    title: "If X and Y are independent, the moment generating function (MGF)"
+    in: "X"
+    pages: "retrieved on 2024-11-08"
+    url: "https://x.com/ProbFact/status/1468264616706859016"
+
+proof_id: "P478"
+shortcut: "mgf-sumind"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ and $Y$ be two [independent](/D/ind) [random variables](/D/rvar) and let $Z = X + Y$. Then, the [moment-generating function](/D/mgf) of $Z$ is given by
+
+$$ \label{eq:mgf-sumind}
+M_Z(t) = M_X(t) \cdot M_Y(t)
+$$
+
+where $M_X(t)$, $M_Y(t)$ and $M_Z(t)$ are the [moment-generating functions](/D/mgf) of $X$, $Y$ and $Z$.
+
+
+**Proof:** The [moment-generating function of a random variable](/D/mgf) $X$ is
+
+$$ \label{eq:mfg}
+M_X(t) = \mathrm{E} \left( \exp \left[ t X \right] \right)
+$$
+
+and therefore the moment-generating function of the sum $Z$ is given by
+
+$$ \label{eq:mgf-sumind-s1}
+\begin{split}
+M_Z(t)
+&= \mathrm{E} \left( \exp \left[ t Z \right] \right) \\
+&= \mathrm{E} \left( \exp \left[ t (X + Y) \right] \right) \\
+&= \mathrm{E} \left( \exp \left[ t X \right] \cdot \exp \left[ t Y \right] \right) \; .
+\end{split}
+$$
+
+Because the [expected value is multiplicative for independent random variables](/P/mean-mult), we have
+
+$$ \label{eq:mgf-sumind-s2}
+\begin{split}
+M_Z(t)
+&= \mathrm{E} \left( \exp \left[ t X \right] \right) \cdot \mathrm{E} \left( \exp \left[ t Y \right] \right) \\
+&= M_X(t) \cdot M_Y(t) \; .
+\end{split}
+$$