The exponential integral distribution (Q1820494)

For a positive integer n let \(E_ n^{(0)}(x)=\exp (-x)\) and \[ E_ n^{(m+1)}(x)=x^{n-1}\int^{\infty}_{x}E_ n^{(m)}(t)t^{- n}dt\quad for\quad m=0,1,2,.... \] The exponential integral distribution with parameters m, n and \(\nu\) \((>0)\) is then described by the probability density function \[ f(x)=(n+\nu -1)^ mx^{\nu -1}E_ n^{(m)}(x)/\Gamma (\nu),\quad for\quad x>0. \] Expressions for the moments and the cumulative distribution function are given and physical relevance of this distribution is discussed. With \(m=0\), this reduces to a gamma distribution.