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Clarify Student's t-distribution Text #28

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6 changes: 4 additions & 2 deletions chapterProbability.tex
Original file line number Diff line number Diff line change
Expand Up @@ -290,7 +290,8 @@ \subsection{Gaussian (normal) distribution}

\subsection{Student's t-distribution}
\begin{table}
\caption{Summary of Student's t-distribution.}
\caption{Summary of Student's t-distribution.}
\label{tab:student-t-dist}
\centering
\begin{tabular}{cccccc}
\hline\noalign{\smallskip}
Expand All @@ -300,6 +301,7 @@ \subsection{Student's t-distribution}
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
Student's $t$-distribution is summarized in Table \ref{tab:student-t-dist}
where $\Gamma(x)$ is the gamma function:
\begin{equation}
\Gamma(x) \triangleq \int_0^\infty t^{x-1}e^{-t}\mathrm{d}t
Expand All @@ -323,7 +325,7 @@ \subsection{Student's t-distribution}
\subfloat[]{\includegraphics[scale=.70]{robustness-a.png}} \\
\subfloat[]{\includegraphics[scale=.70]{robustness-b.png}}
\caption{Illustration of the effect of outliers on fitting Gaussian, Student and Laplace distributions. (a) No outliers (the Gaussian and Student curves are on top of each other). (b) With outliers. We see that the Gaussian is more affected by outliers than the Student and Laplace distributions.}
\label{fig:robustness}
\label{fig:robustness}
\end{figure}

If $\nu=1$, this distribution is known as the \textbf{Cauchy} or \textbf{Lorentz} distribution. This is notable for having such heavy tails that the integral that defines the mean does not converge.
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Binary file modified machine-learning-cheat-sheet.pdf
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