tex: add more NFBP graphs

Probas.py

@@ -368,6 +368,6 @@ def basic_demo():
         )
 
 
-stats_NFBP_iter(10**3, 10)
+stats_NFBP_iter(10**5, 50)
 print("\n\n")
 stats_NFDBP(10**3, 10, 1)

latex/content.tex

@@ -243,41 +243,40 @@ is the same as for $ T_i $, yielding $ \overline{V_1} = 0.897 $, $ {S_N}^2 =
 the two values for $ V_1 $ are high (being bouded between $ 0 $ and $ 1 $).
 
 
-\paragraph{1 000 000 simulations} In order to ensure better precision, we then
+\paragraph{100 000 simulations} In order to ensure better precision, we then
-ran $ R = 10^6 $ simulations with $ N	= 50 $ different items each.
+ran $ R = 10^5 $ simulations with $ N	= 50 $ different items each.
 
-With 10 6 simulations, we obtain Xn barre = cf graphe
+\begin{figure}[h]
-Calcul Sn carre
+	\centering
-IC observed
+	\includegraphics[width=0.8\textwidth]{graphics/graphic-NFBP-Ti-105-sim}
+	\caption{Histogram of $ T_i $ for $ R = 10^5 $ simulations and $ N = 50 $ items (number of items per bin)}
+	\label{fig:graphic-NFBP-Ti-105-sim}
+\end{figure}
 
+On this graph (figure \ref{fig:graphic-NFBP-Ti-2-sim}), we can see each value
+of $ T_i $. Our calculations have yielded that $ \overline{T_1} = 1.72 $ and $
+	{S_N}^2 = 0.88 $. Our Student coefficient is $ t_{0.95, 2} = 2 $.
 
-Same for V.
+We can now calculate the Confidence Interval for $ T_1 $ for $ R = 10^5 $ simulations :
 
+\begin{align*}
+	IC_{95\%}(T_1) & = \left[ 1.72 \pm 1.96 \frac{\sqrt{0.88}}{\sqrt{10^5}} \cdot 2 \right] \\
+	               & = \left[ 172 \pm 0.012 \right]                                         \\
+\end{align*}
 
-Graphe H
+We can see that the Confidence Interval is very small, thanks to the large number of iterations.
+This results in a steady curve in figure \ref{fig:graphic-NFBP-Ti-105-sim}.
 
-\paragraph{Distribution of $ T_i $} We first studied how many items were
+\begin{figure}[h]
-present per bin.
+	\centering
-
+	\includegraphics[width=0.8\textwidth]{graphics/graphic-NFBP-Vi-105-sim}
-% TODO sim of T_i
+	\caption{Histogram of $ V_i $ for $ R = 10^5 $ simulations and $ N = 50 $ items (size of the first item in a bin)}
-
+	\label{fig:graphic-NFBP-Vi-105-sim}
-We determined the empirical mean to be
+\end{figure}
-
-\[
-	\overline{T_i} = \frac{1}{20} \sum_{k=1}^{20} T_k = 1.5 \qquad \forall 1 \leq i \leq 20
-\]
-
-
-We can show
-
-\paragraph{Distribution of $ V_i $} We then looked at the size of the first
-item in each bin.
 
 \paragraph{Asymptotic behavior of $ H_n $} Finally, we analyzed how many bins
-were needed to store $ n $ items.
+were needed to store $ n $ items. We used the numbers from the $ R = 10^5 $ simulations.
 
-% TODO histograms
-% TODO analysis histograms