From bbe25c3d4cf3fae22ee41f4e9f32a4b48dd5f5e8 Mon Sep 17 00:00:00 2001
From: Paul ALNET <paul@shimaore.net>
Date: Sun, 4 Jun 2023 22:16:30 +0200
Subject: [PATCH] tex: NFDBP add expected value T_i

---
 latex/content.tex | 23 +++++++++++++++++++++++
 1 file changed, 23 insertions(+)

diff --git a/latex/content.tex b/latex/content.tex
index 5699d70..2bca0ca 100644
--- a/latex/content.tex
+++ b/latex/content.tex
@@ -396,9 +396,32 @@ Finally, plugging this into \eqref{eq:prob} gives us
 	P(T_i = k) = P(A_{k-1}) - P(A_{k}) = \frac{1}{(k-1)!} - \frac{1}{k!} \qquad \forall k \geq 2
 \]
 
+\subsection{Expected value of $ T_i $}
 
+We now compute the expected value $ \mu $ and variance $ \sigma^2 $ of $ T_i $.
 
+\begin{align*}
+	\mu = E(T_i) & = \sum_{k=2}^{\infty} k \cdot P(T_i = k)                    \\
+	             & = \sum_{k=2}^{\infty} (\frac{k}{(k-1)!} - \frac{1}{(k-1)!}) \\
+	             & = \sum_{k=2}^{\infty} \frac{k-1}{(k-1)!}                    \\
+	             & = \sum_{k=0}^{\infty} \frac{1}{k!}                          \\
+	             & = e                                                         \\
+\end{align*}
 
+\begin{align*}
+	E({T_i}^2) & = \sum_{k=2}^{\infty} k^2 \cdot P(T_i = k)                    \\
+	           & = \sum_{k=2}^{\infty} (\frac{k^2}{(k-1)!} - \frac{k}{(k-1)!}) \\
+	           & = \sum_{k=2}^{\infty} \frac{(k-1)k}{(k-1)!}                   \\
+	           & = \sum_{k=2}^{\infty} \frac{k}{(k-2)!}                        \\
+	           & = \sum_{k=0}^{\infty} \frac{k+2}{k!}                          \\
+	           & = \sum_{k=0}^{\infty} (\frac{1}{(k-1)!} + \frac{2}{(k)!})     \\
+	           & = \sum_{k=0}^{\infty} \frac{1}{(k)!} - 1 + 2e                 \\
+	           & = 3e - 1
+\end{align*}
+
+\begin{align*}
+	\sigma^2 = E({T_i}^2) - E(T_i)^2 = 3e - 1 - e^2
+\end{align*}
 
 
 \section{Complexity and implementation optimization}