MedianOfADistribution 2004-06-07 21:14:30 2009-02-05 21:11:17 Definition median % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given a probability distribution (density) function $f_X(x)$ on $\Omega$ over a random variable $X$, with the associated probability measure $P$, a \emph{median} $m$ of $f_X$ is a real number such that \begin{enumerate} \item $P(X\leq m)\geq \frac{1}{2},$ \item $P(X\geq m)\geq \frac{1}{2}.$ \end{enumerate} The median is also known as the $50^{\text{th}}$-percentile or the second quartile. \textbf{Examples:} \begin{itemize} \item An example from a discrete distribution. Let $\Omega=\mathbb{R}$. Suppose the random variable $X$ has the following distribution: $P(X=0)=0.99$ and $P(X=1000)=0.01$. Then we can easily see the median is 0. \item Another example from a discrete distribution. Again, let $\Omega=\mathbb{R}$. Suppose the random variable $X$ has distribution $P(X=0)=0.5$ and $P(X=1000)=0.5$. Then we see that the median is not unique. In fact, all real values in the interval $[0,1000]$ are medians. \item In practice, however, the median may be calculated as follows: if there are $N$ numeric data points, then by ordering the data values (either non-decreasingly or non-increasingly), \begin{enumerate} \item the $(\frac{N+1}{2})$-th data point is the median if $N$ is odd, and \item the midpoint of the $(N-1)$th and the $(N+1)$th data points is the median if $N$ is even. \end{enumerate} \item The median of a normal distribution (with mean $\mu$ and variance $\sigma^2$) is $\mu$. In fact, for a normal distribution, mean = median = mode. \item The median of a uniform distribution in the interval $[a,b]$ is $(a+b)/2$. \item The median of a Cauchy distribution with location parameter t and scale parameter s is the location parameter. \item The median of an exponential distribution with location parameter $\mu$ and scale parameter $\beta$ is the scale parameter times the natural log of 2, $\beta\operatorname{ln}2$. \item The median of a Weibull distribution with shape parameter $\gamma$, location parameter $\mu$, and scale parameter $\alpha$ is $\alpha(\operatorname{ln}2)^{1/\gamma}+\mu$. \end{itemize}