FDistribution 2004-06-25 01:38:09 2006-09-26 14:40:59 Definition non-central F distribution % 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 Let $X$ and $Y$ be random variables such that \begin{enumerate} \item $X$ and $Y$ are independent \item $X\sim \chi^2(m)$, the \PMlinkname{chi-squared distribution}{ChiSquaredRandomVariable} with $m$ degrees of freedom \item $Y\sim \chi^2(n)$, the chi-squared distribution with $n$ degrees of freedom \end{enumerate} Define a new random variable $Z$ by $$Z=\frac{(X/m)}{(Y/n)}.$$ Then the distribution of $Z$ is called the \emph{central F distribution}, or simply the \emph{F distribution with m and n degrees of freedom}, denoted by $Z\sim \operatorname{F}(m,n)$. By transformation of the random variables $X$ and $Y$, one can show that the probability density function of the F distribution of $Z$ has the form: $$f_Z(x)=\frac{m^{m/2}n^{n/2}}{\operatorname{B}(\frac{m}{2},\frac{n}{2})} \cdot\frac{x^{(m/2)-1}}{(mx+n)^{(m+n)/2}},$$ for $x>0$, where $\operatorname{B}(\alpha,\beta)$ is the beta function. $f_Z(x)=0$ for $x\le 0$. For a fixed $m$, say 10, below are some graphs for the probability density functions of the F distribution with $(m,n)$ degrees of freedom. \begin{center} \includegraphics[scale=0.9]{fdist1} \end{center} The next set of graphs shows the density functions with $(m,n)$ degrees of freedom when $n$ is fixed. In this example, $n=10$. \begin{center} \includegraphics[scale=0.9]{fdist2} \end{center} If $X\sim \chi^2(m,\lambda)$, the non-central chi-square distribution with m degrees of freedom and non-centrality parameter $\lambda$, with $Y$ and $Z$ defined as above, then the distribution of $Z$ is called the \emph{non-central F distribution with m and n degrees of freedom and non-centrality parameter} $\lambda$. \textbf{Remarks} \begin{itemize} \item the ``F'' in the F distribution is given in honor of statistician R. A. Fisher. \item If $X\sim \operatorname{F}(m,n)$, then $1/X\sim \operatorname{F}(n,m)$. \item If $X\sim \operatorname{t}(n)$, the t distribution with $n$ degrees of freedom, then $X^2\sim \operatorname{F}(1,n)$. \item If $X\sim \operatorname{F}(m,n)$, then $$\operatorname{E}[X] = \frac{n}{n-2}\mbox{ if }n>2,$$ and $$\operatorname{Var}[X] = \frac{2n^2(m+n-2)}{m(n-2)^2(n-4)}\mbox{ if }n>4.$$ \item Suppose $X_1,\ldots,X_m$ are random samples from a normal distribution with mean $\mu_1$ and variance $\sigma_1^2$. Furthermore, suppose $Y_1,\ldots,Y_n$ are random samples from another normal distribution with mean $\mu_2$ and variance $\sigma_2^2$. Then the statistic defined by $$V=\frac{\hat{\sigma_1}^2}{\hat{\sigma_2}^2},$$ where $\hat{\sigma_1}^2$ and $\hat{\sigma_1}^2$ are sample variances of the $X_i's$ and the $Y_j's$, respectively, has an F distribution with m and n degrees of freedom. $V$ can be used to test whether $\sigma_1^2=\sigma_2^2$. $V$ is an example of an F test. \end{itemize}