📂Probability Distribution

Mean and Variance of the F-distribution

Formulas

$X \sim F ( r_{1} , r_{2})$ Surface Area $$E(X) = {{ r_{2} } \over { r_{2} - 2 }} \qquad , r_{2} > 2 \\ \Var(X) = {{ 2 d_{2}^{2} (d_{1} + d_{2} - 2) } \over { d_{1} (d_{2} -2)^{2} (d_{2} - 4) }} \qquad , r_{2} > 4$$

Derivation

Strategy: Like the chi-squared distribution, the F-distribution also has known moment-generating functions, which we will use.

Moments of the F-Distribution: Let’s say $X \sim F(r_{1} , r_{2})$ and it can be expressed as $\displaystyle X = {{ X_{1} } \over { X_{2} }}$. If $X_{1}$ and $X_{2}$ both follow chi-squared distributions with degrees of freedom $d_{1}, d_{2}$ and if $d_{2} > 2k$ then the $k$-th moment exists $$EX^{k} = \left( {{ r_{2} } \over { r_{1} }} \right)^{k} E X_{1}^{k} E X_{2}^{-k}$$

Moments of the Chi-Squared Distribution: Let’s say $X \sim \chi^{2} (r)$. If $k > - r/ 2$ then the $k$-th moment exists $$E X^{k} = {{ 2^{k} \Gamma (r/2 + k) } \over { \Gamma (r/2) }}$$

Mean

Assuming $r_{2} > 2$ then because $-k = -1 > -r_{2} / 2$ thus, $EX_{2}^{-1}$ exists.

If $k=1$ then according to the moment-generating functions $$\begin{align*} EX^{1} =& \left( {{ r_{2} } \over { r_{1} }} \right)^{1} E X_{1}^{1} E X_{2}^{-1} \\ =& {{ r_{2} } \over { r_{1} }} {{ 2^{1} \Gamma (r_{1}/2 + 1) } \over { \Gamma (r_{1}/2) }} {{ 2^{-1} \Gamma (r_{2}/2 -1 ) } \over { \Gamma (r_{2}/2) }} \\ =& {{ r_{2} } \over { r_{1} }} 2r_{1} {{ 1 } \over { r_{2}/2 - 1 }} \\ =& {{ r_{2} } \over { r_{2} - 2 }} \end{align*}$$

■

Variance

Assuming $r_{2} > 4$ then because $-k = -2 > -r_{2} /2$ thus, $E X_{2}^{-2}$ exists.

If $k=2$ then according to the moment-generating functions $$\begin{align*} EX^{2} =& \left( {{ r_{2} } \over { r_{1} }} \right)^{2} E X_{1}^{2} E X_{2}^{-2} \\ =& \left( {{ r_{2} } \over { r_{1} }} \right)^{2} {{ 2^{2} \Gamma (r_{1}/2 + 2) } \over { \Gamma (r_{1}/2) }} {{ 2^{-2} \Gamma (r_{2}/2 -2 ) } \over { \Gamma (r_{2}/2) }} \\ =& \left( {{ r_{2} } \over { r_{1} }} \right)^{2} {{ (r_{1}/2+1)r_{1}/2 } \over { (r_{2}/2-1) (r_{2}/2-2) }} \\ =& \left( {{ r_{2} } \over { r_{1} }} \right)^{2} {{ (r_{1}+2)r_{1} } \over { (r_{2}-2) (r_{2}-4) }} \\ =& {{ r_{2}^{2} (r_{1}+2) } \over { r_{1} (r_{2}-2) (r_{2}-4) }} \end{align*}$$ Therefore $$\begin{align*} \Var(X) =& {{ r_{2}^{2} (r_{1}+2) } \over { r_{1} (r_{2}-2) (r_{2}-4) }} - \left[ {{ r_{2} } \over { r_{2} - 2 }} \right]^{2} \\ =& {{ r_{2}^{2} } \over { r_{1} (r_{2} -2)^{2} (r_{2} - 4) }} \left[ (r_{1} + 2)(r_{2} - 2) - r_{1}(r_{2} - 4) \right] \\ =& {{ 2 r_{2}^{2} (r_{1} + r_{2} - 2) } \over { r_{1} (r_{2} -2)^{2} (r_{2} - 4) }} \end{align*}$$

■