F, T, Chi, N, Gamma

normal distribution

t distribution

chi-square distribution

may be seen as special cases of the F distribution:

F-distribution is a special case of type 6 Pearson distribution.

normal distribution = F(1,infinite)
t^2 distribution = F(1, n2)
chi-square distribution = F(n1, infinite)

http://www.statlect.com/subon2/ucdchi1.htm

Differential equation

The pdf of the chi-squared distribution is a solution to the following differential equation:

\left\{\begin{array}{l} 2 x f'(x)+f(x) (-k+x+2)=0, \\ f(1)=\frac{2^{-k/2}}{\sqrt{e} \Gamma \left(\frac{k}{2}\right)} \end{array}\right\}

Differential equation

The probability density function of the F-distribution is a solution of the following differential equation:

\left\{\begin{array}{l} 2 x \left(d_1 x+d_2\right) f'(x)+\left(2 d_1 x+d_2 d_1 x-d_2 d_1+2 d_2\right) f(x)=0, \\[12pt] f(1)=\frac{d_1^{\frac{d_1}{2}} d_2^{\frac{d_2}{2}} \left(d_1+d_2\right){}^{\frac{1}{2} \left(-d_1-d_2\right)}}{B\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \end{array}\right\}

Differential equation

The pdf of the t-distribution is a solution to the following differential equation:

\left\{\begin{array}{l} \left(\nu+x^2\right) f'(x)+(\nu +1) x f(x)=0, \\ f(1)=\frac{\nu^{\nu/2} (\nu +1)^{-\frac{\nu}{2}-\frac{1}{2}}}{B\left(\frac{\nu}{2}, \frac{1}{2}\right)} \end{array}\right\}

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