# F distribution

Let $X$ and $Y$ be random variables such that

1. 1.

$X$ and $Y$ are independent

2. 2.

$X\sim\chi^{2}(m)$, the chi-squared distribution (http://planetmath.org/ChiSquaredRandomVariable) with $m$ degrees of freedom

3. 3.

$Y\sim\chi^{2}(n)$, the chi-squared distribution with $n$ degrees of freedom

Define a new random variable $Z$ by

 $Z=\frac{(X/m)}{(Y/n)}.$

Then the distribution of $Z$ is called the central F distribution, or simply the 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 $x\leq 0$.

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.

The next set of graphs shows the density functions with $(m,n)$ degrees of freedom when $n$ is fixed. In this example, $n=10$.

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 non-central F distribution with m and n degrees of freedom and non-centrality parameter $\lambda$.

Remarks

• the “F” in the F distribution is given in honor of statistician R. A. Fisher.

• If $X\sim\operatorname{F}(m,n)$, then $1/X\sim\operatorname{F}(n,m)$.

• If $X\sim\operatorname{t}(n)$, the t distribution with $n$ degrees of freedom, then $X^{2}\sim\operatorname{F}(1,n)$.

• If $X\sim\operatorname{F}(m,n)$, then

 $\operatorname{E}[X]=\frac{n}{n-2}\mbox{ if }n>2,$

and

 $\operatorname{Var}[X]=\frac{2n^{2}(m+n-2)}{m(n-2)^{2}(n-4)}\mbox{ if }n>4.$
• 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 $X_{i}^{\prime}s$ and the $Y_{j}^{\prime}s$, 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.

Title F distribution FDistribution 2013-03-22 14:26:56 2013-03-22 14:26:56 CWoo (3771) CWoo (3771) 15 CWoo (3771) Definition msc 62A01 Fisher F distribution F-distribution central F-distribution central F distribution non-central F distribution