Day 11 : F

Day 11 : F#

The F-distribution is also known as F-ratio, Snedecor’s F distribution, or the Fisher–Snedecor distribution after British polymath Ronald Fisher and American mathematician George W. Snedecor. It is a continuous probability distribution with support on \([0, \infty)\), defined by two parameters \(d_1, d_2 >0\) which are called degrees of freedom. More precisely, the \(F\) distribution with \(d_1\) and \(d_2\), degrees of freedom is the distribution given by \(X = \frac{S_1/ d_1}{ S_2/d_2}\) where \(S_1\) and \(S_2\) are independent random variables with chi-square distributions with respective degrees of freedom \(d_1\) and \(d_2\).

The probability density function is given by

\[f(x) = \frac{1}{x B\left(\frac{d_1}{2}, \frac{d_2}{2} \right)} \sqrt{ \frac{ (d_1x)^{d1} d_2^{d2} }{ (d_1x + d_2)^{d_1 + d_2} } }\]

where \(B\) is the beta function.

The cumulative distribution function is given by

\[F(x) = I_{d_1x/(d_1x + d_2)}\left( \frac{d_1}{2}, \frac{d_2}{2} \right)\]

where \(I\) is the regularized incomplete beta function.

🔔 Random Facts 🔔#