एरलांग वितरण

Erlang

Probability density function
Cumulative distribution function
Parameters	$k\in \{1,2,3,\ldots \},$ shape $\lambda \in (0,\infty ),$ rate alt.: $\beta =1/\lambda ,$ scale
Support	$x\in [0,\infty )$
PDF	${\frac {\lambda ^{k}x^{k-1}e^{-\lambda x}}{(k-1)!}}$
CDF	$P(k,\lambda x)={\frac {\gamma (k,\lambda x)}{(k-1)!}}=1-\sum _{n=0}^{k-1}{\frac {1}{n!}}e^{-\lambda x}(\lambda x)^{n}$
Mean	${\frac {k}{\lambda }}$
Median	No simple closed form
Mode	${\frac {1}{\lambda }}(k-1)$
Variance	${\frac {k}{\lambda ^{2}}}$
Skewness	${\frac {2}{\sqrt {k}}}$
Ex. kurtosis	${\frac {6}{k}}$
Entropy	$(1-k)\psi (k)+\ln \left[{\frac {\Gamma (k)}{\lambda }}\right]+k$
MGF	$\left(1-{\frac {t}{\lambda }}\right)^{-k}$ for $t<\lambda$
CF	${(1-<!--\; -\; -->\frac{it}{\mathrm{\lambda <!--\; \lambda \; -->}}}^{}$