tex: NFBP stats

latex/content.tex

@@ -180,14 +180,75 @@ Mathematically, the NFBP algorithm imposes the following constraint on the first
 We implemented the NFBP algorithm in Python \footnotemark, for its ease of use
 and broad recommendation. We used the \texttt{random} library to generate
 random numbers between $ 0 $ and $ 1 $ and \texttt{matplotlib} to plot the
-results in the form of histograms. We ran $ R = 10^6 $ simulations with
-$ N = 10 $ different items each.
+results in the form of histograms.
 
 \footnotetext{The code is available in Annex \ref{annex:probabilistic}}
 
+We will try to approximate $ \mathbb{E}[X] $ and $ \mathbb{E}[V] $ with $
+	\overline{X_N} $ using $ {S_n}^2 $. This operation will be done for both $ R =
+	2 $ and $ R = 10^6 $ simulations.
+
+\[
+	\overline{X_N} = \frac{1}{N} \sum_{i=1}^{N} X_i
+\]
+
+As the variance value is unknown, we will use $ {S_n}^2 $ to estimate the
+variance and further determine the Confidence Interval (95 \% certainty).
+
+\begin{align*}
+	{S_N}^2      & = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \overline{X_N})^2                                      \\
+	IC_{95\%}(m) & = \left[ \overline{X_N} \pm \frac{S_N}{\sqrt{N}} \cdot t_{1 - \frac{\alpha}{2}, N-1} \right] \\
+\end{align*}
+
+
+
+
+\paragraph{2 simulations} We first ran $ R = 2 $ simulations to observe the
+behavior of the algorithm and the low precision of the results.
+
+% TODO graph T_i 2 sim
+
+On this graph, we can see each value of $ T_i $. Our calculations have yielded
+that $ \overline{T_1} = 1.0 $ and $ {S_N}^2 = 2.7 $. Our student coefficient is
+$ t_{0.95, 2} = 4.303 $.
+
+\begin{align*}
+	\overline{T_1} = \sum_{k=1}^{2} {T_1}_k & = 1.0                                                                 \\
+	IC_{95\%}(T_1)                          & = \left[ 1.0 \pm 1.96 \frac{\sqrt{2.7}}{\sqrt{2}} \cdot 4.303 \right] \\
+	                                        & = \left[ 1 \pm 9.8 \right]                                            \\
+\end{align*}
+
+With two simulations, we obtain $ \overline{T_1} = 1.0 $.
+
+
+
+IC observed
+
+We then ran $ R = 10^6 $ simulations with $ N = 50 $ different items each.
+With 10 6 simulations, we obtain Xn barre = cf graphe
+Calcul Sn carre
+IC observed
+
+
+Same for V.
+
+
+Graphe H
+
 \paragraph{Distribution of $ T_i $} We first studied how many items were
 present per bin.
 
+% TODO sim of T_i
+
+We determined the empirical mean to be
+
+\[
+	\overline{T_i} = \frac{1}{20} \sum_{k=1}^{20} T_k = 1.5 \qquad \forall 1 \leq i \leq 20
+\]
+
+
+We can show
+
 \paragraph{Distribution of $ V_i $} We then looked at the size of the first
 item in each bin.