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  "body": "\n## Definition\n\nLet $d_1$ and $d_2$ be positive integers.\nLet $U_1$ and $U_2$ be [[independent|0E9502NT92]] random variables with [[chi-squared distributions|0RFMMBHFTV]] with $d_1$ and $d_2$ degrees of freedom.\nThe distribution of\n$$X = \\frac{U_1 / d_1}{U_2 / d_2}$$\nis the *F-distribution* with $d_1$ and $d_2$ degrees of freedom, written $F(d_1, d_2)$.\n\n## Remarks\n\nIt has the [[probability density function|04YG690N5V]]\n$$f(x) = \\frac{1}{B(d_1/2, d_2/2)} \\left(\\frac{d_1}{d_2}\\right)^{d_1/2} x^{d_1/2 - 1} \\left(1 + \\frac{d_1}{d_2} x\\right)^{-(d_1 + d_2)/2}, \\quad x \u003e 0,$$\nwhere $B$ is the beta function.\nThe same formula with real $d_1, d_2 \u003e 0$ defines $F(d_1, d_2)$ for non-integer parameters.\nIts mean is $d_2/(d_2 - 2)$ when $d_2 \u003e 2$.\nIf $X \\sim F(d_1, d_2)$, then $1/X \\sim F(d_2, d_1)$.\n\nExample: let $T$ have the [[Student t-distribution|0NQ5798SB1]] with $\\nu$ degrees of freedom, so $T = Z/\\sqrt{V/\\nu}$ with $Z$ standard normal and $V$ chi-squared with $\\nu$ degrees of freedom, independent.\nThen $T^2 = (Z^2/1)/(V/\\nu)$, and $Z^2$ is chi-squared with $1$ degree of freedom, so $T^2 \\sim F(1, \\nu)$.\n\nNon-example: the independence clause matters.\nIf $d_1 = d_2$ and $U_2 = U_1$, then $X = 1$ always, which is not an F-distribution.\n",
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