Statistics - F distribution



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The F distribution (Snedecor's F distribution or the Fisher–Snedecor distribution) represents continuous probability distribution which occurs frequently as null distribution of test statistics. It happens mostly during analysis of variance or F-test.

F Distribution

Probability density function

Probability density function of F distribution is given as:

Formula

${ f(x; d_1, d_2) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1x+d_2)^{d_1+d_2}}}}{x \beta (\frac{d_1}{2}, \frac{d_2}{2})} }$

Where −

  • ${d_1}$ = positive parameter.

  • ${d_2}$ = positive parameter.

  • ${x}$ = random variable.

Cumulative distribution function

Cumulative distribution function of F distribution is given as:

Formula

${ F(x; d_1, d_2) = I_{\frac{d_1x}{d_1x+d_2}}(\frac{d_1}{2}, \frac{d_2}{2})}$

Where −

  • ${d_1}$ = positive parameter.

  • ${d_2}$ = positive parameter.

  • ${x}$ = random variable.

  • ${I} $ = lower incomplete beta function.



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