This is a quick reference of all Burr type distributions with their CDF, PDF and PPF functions. I found no complete reference with both of these on the internet, but had to go through papers and multiple websites to compile this.
I have calculated the PDF and PPF functions using FriCAS and Maxima where needed.
Hopefully I didn’t made a mistake typing those in. If you see something wrong or would like to add something drop me a line jan@jadelsbach.de.
Burr Type 2
CDF: $$ F(x)=(1+e^{-x})^{-p} $$ \( \forall x \in 0 < x < 1 \)
PDF: $$ f(x)=pe^{-x}(e^{-x}+1)^{-p-1} $$
PPF: $$ f(k)=-\log\left(k^{-1/p}-1\right) $$
Burr Type 3 (Dagum)
CDF: $$ F(x)=(1+x^{-a})^{-p} $$ \( \forall x \in 0 < x < \infty \)
PDF: $$ f(x)=apx^{-a-1}(x^{-a}+1)^{-p-1} $$
PPF: $$ f(k)=(k^{-1/p}-1)^{-1/a} $$
Burr Type 4
CDF: $$ F(x)=\frac{x}{1+(p^{-1/r}-1)^c} $$ \( \forall x \in 0 < x < c \)
PDF: $$ f(x)=\frac{1}{1+(p^{-1/r}-1)^c} $$
PPF: $$ f(k)=k\left(p^{-1/r}-1\right)^x+k $$
Burr Type 5
CDF: $$ F(x)=(1+ce^{-\tan(x)})^{-q} $$ \( \forall x \in -\pi/2<x<\pi/2 \)
PDF: $$ f(x)=(cq\tan(x)^2+cq)e^{-\tan(x)}(ce^{-\tan(x)}+1)^{-q-1} $$
PPF: $$ f(k)=-\tan^{-1}\left[\log\left(\frac{e^{-\log(k)/q}-1}{c}\right)\right] $$
Burr Type 6
CDF: $$ F(x)=(1+e^{-x\sinh(x)})^{-q} $$ \( \forall x \in \mathbb{R} \)
PDF: $$ f(x)=(q\sinh(x)+qx\cosh(x))e^{-x\sinh(x)}(e^{-x\sinh(x)}+1)^{-q-1} $$
PPF: $$ f(k)=\frac{e^{qx\sinh(x)}}{\left(e^{x\sinh(x)}+1\right)^q} $$
Burr Type 7
CDF: $$ F(x)=2^{-q}(1+\tanh(x))^q $$ \( \forall x \in \mathbb{R} \)
PDF: $$ f(x)=2^{-q}(-q\tanh(x)^2+q)(\tanh(x)+1)^{q-1} $$
PPF: $$ f(k)=\tanh^{-1}\left(2k^{1/q}-1\right) $$
Burr Type 8
CDF: $$ F(x)=\left(\frac{2}{\pi}\tan^{-1}(e^x)\right)^q $$ \( \forall x \in \mathbb{R} \)
PDF: $$ f(x) = \frac{2qe^x\left(\frac{2\tan^{-1}(e^x)}{\pi}\right)^{q-1}}{\pi e^{2x}+\pi} $$
PPF: $$ f(k)=\log\left[\tan\left(\frac{(k\pi^q)^{1/q}}{2}\right)\right] $$
Burr Type 9
CDF: $$ F(x)=1-\frac{2}{2+c((1+e^x)^q-1)} $$ \( \forall x \in \mathbb{R} \)
PDF: $$ f(x) = \frac{2cqe^x(e^x+1)^{q-1}}{c^2((e^x+1)^q)^2+(-2c^2+4c)(e^x+1)^q+c^2-4c+4} $$
PPF: $$ f(k)=\log\left(e^{\log\left(\frac{(c-2)k-c}{ck-c}\right)/q}-1\right) $$
Burr Type 10
CDF: $$ F(x)=(1-e^{-x^2})^a $$ \( \forall x \in \mathbb{R} \)
PDF: $$ f(x)=2axe^{-ax^2} $$
PPF: $$ f(k)=\sqrt{-\log\left(-e^{\log(k)/a}+1\right)} $$
Burr Type 11
CDF: $$ F(x)=\left(x-\frac{1}{2\pi}\sin(2\pi x)\right)^q $$ \( \forall x \in \ 0< x < 1 \)
PDF: $$ f(x)=q(1-\cos(2\pi x))\left(x-\frac{\sin(2\pi x)}{2\pi}\right)^{q-1} $$
PPF: $$ f(k)=\left(x-\frac{\sin(2\pi x)}{2\pi}\right)^q $$
Burr Type 12 (Singh-Maddala)
CDF: $$ F(x)=1-(1+x^a)^{-q} $$ \( \forall x \in \ 0\le x < \infty \)
PDF: $$ f(x)=aqx^{a-1}(1+x^a)^{-q-1} $$
PPF: $$ f(k)=e^{\frac{\log(-\log(-k+1)/q)-1}{a}} $$