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F Distribution

F-distributions arise frequently as null distributions of test statistics, most notably in the analysis of variance.

What is F-distribution?

The F is another commonly encountered distribution when testing hypotheses about model parameters. The F-distribution has two parameters,v1 and v2, respectively known as the numerator and denominator degrees of freedom. These parameters are usually subscripts in the shorthand expression Fv1,v2.
F-distribution is the ratio of two independent \ two random variables where each has been divided by its degree of freedom.
$ Y = \frac{\frac{W1}{v1}}{\frac{W2}{v2}} $
If Y~Fv1,v2, then the mean of Y is $ E[Y] = \frac{v2}{v2-2} $ 

The variance is given by:
$ \frac{{2v^{2}_2}\left ( v_{1}+v_{2}-2 \right )}{v_{1}\left ( v_{2}-2 \right )^{2} \left (v_{2}-4 \right )} $

F Distribution :
$ f_{F}\left ( x|V_{1},V_{2} \right )
 =
\frac{}{}\frac{\Gamma\left ( \frac{V_{1}+V_{2}}{2} \right )}{\Gamma\left ( \frac{V_{1}}{2 } \right )\Gamma\left ( \frac{V_{2}}{2 } \right )}\left (\frac{V_{1}}{V_{2}} \right )^{\frac{v_{1}}{2}}
\frac{x^{\frac{v_{1}-2}{2}}}{\left ( 1+ \left ( \frac{V_{1}}{V_{2}} \right )x
\right )^{{\frac{v_{1}+v_{2}}{2}}}
} $

Why is F-distribution important?

The main application of the F-distribution is to see if two independent samples for normal populations with the same variance have been drawn or if two independent estimates of the population variance are homogenous or not, as it is typically more helpful to compare two variances than two averages. It is useful when analysts are trying to compare distributions of two different stocks.

Owais Siddiqui
1 min read
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