\chapter{Introduction}
	\section{Section name}
		\subsection{subsection name}
	\section{Background and Rationale} 
	
	\section{Introduction to the Mixture Distribution}
	\indent Simply speaking, a mixture distribution is a probability distribution which consists 2 or more distributions where they mix in a way such that, area under the final density curve is equal to 1. More formally, we say that a distribution $f$ given by equation \ref{eq:mixpdf} is a mixture of $K$ component distributions $f_1, f_2, ..., f_K$ if 
	\begin{equation}
	\label{eq:mixpdf}
	f(x) =  \sum_{k=1}^K \lambda_{k}f_{k}(x)
	\end{equation}
	with $\lambda_{k}$ being the mixing weights $\lambda_{k} > 0$, $\sum_{k} \lambda_{k} = 1$ \citep{picard2007introduction}.
	  
	\indent In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution \citep{intro}.
	\\\indent The distributions can be made up of different distributions (e.g. a normal distribution and a t-distribution) or they can be made up of the same distribution with different parameters. For example, the following image (Figure \ref{fig:multimodel}) shows a mixture of three normal distributions (called a Gaussian Mixture Model), each with a different mean, but equal weight of 1/3 and equal variance.
	\begin{figure}[H]
		\includegraphics{./Figures/multimodel.PNG}
		\caption{\label{fig:multimodel}A mixture of three normal distributions with equal variances and with means 5, 10, and 15}
	\end{figure}
	
	\section{Objectives of the Study}
	The objectives of the study are as follows:
	\begin{enumerate}
		\item Perform a comparison among several non-parametric density estimation methods utilizing literature.
		\item Evaluate the performance of the Kernel Density Estimator for various existing bandwidth selection methods and kernel functions for various Gaussian mixtures.
		\item Investigate on the possibility of improving the practical performance of the bandwidth selector to devise a more efficient bandwidth selection method. 	
		\item Apply proposed estimation methods to estimate average adult human height distributions.
		\item Analyze the distributions to identify modes of female and male adult height in various regions and how it has changed over the years (generations). 
	\end{enumerate}