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Density Estimation

Density estimation is a method of obtaining information on the distribution of an observed population by constructing an estimate on the data’s density function. The probability density function which is estimated represents the population of interest. The data underneath the function represents a random sample from the population.

There are both parametric and nonparametric approaches to constructing the density estimate. Parametric estimates are based from data of the parametric family of distributions, such as the normal distribution1. Nonparametric density estimates have less rigid assumptions associated with them and the data in this case is still assumed to have a probability density1.

There are two common ways of carrying out density estimation1:

  1. Using a histogram
  2. Kernel density estimation

Histograms represent a visual way of estimating the attributes and characteristics of a data set. The shape and spread of a histogram provide an idea of the data’s behaviour. Although, there are disadvantages associated with using histograms for density estimation. Histograms are dependent upon the width, as well as the start and ending points of their plotted bars, and thus, histograms do not provide a smooth estimate1. They focus on end points and not on the centre of each data point in the distribution.

On the other hand, Kernel estimators do consider the centre of each point in the data set and provide a smooth density estimate. In the Kernel estimate, the contribution of each data point over the entire set of data points is considered. The Kernel estimate provides a smooth density estimation, whereas the histogram is associated with discreteness.

In statistics, density estimation is important for a multitude of reasons. Not only does it provide an estimate for the distribution of a population, but it provides detail on the structure of a data set. 



1. B.W. Silverman. (2002). Density Estimation for Statistics and Data Analysis. Retrieved from
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Conditional Density Function

Find the conditional density functions for the following experiments. a) a number x is chosen at random in the interval [0,1], given that x>1/4 b) two numbers x and y are chosen at random in the interval [0,1], given that x>y