\mu
\theta
k
The probability density function (pdf) of the normal-exponential-gamma distribution is proportional to
f(x;\mu,k,\theta)\propto\exp{\left(
| (x-\mu)2 | |
| 4\theta2 |
\right)}D-2k-1\left(
| |x-\mu| | |
| \theta |
\right)
where D is a parabolic cylinder function.[1]
As for the Laplace distribution, the pdf of the NEG distribution can be expressed as a mixture of normal distributions,
f(x;\mu,
| infty N(x| | |
| k,\theta)=\int | |
| 0 |
\mu,\sigma2)Exp(\sigma2|\psi)Gamma(\psi|k,1/\theta2)d\sigma2d\psi,
where, in this notation, the distribution-names should be interpreted as meaning the density functions of those distributions.
Within this scale mixture, the scale's mixing distribution (an exponential with a gamma-distributed rate) actually is a Lomax distribution.
The distribution has heavy tails and a sharp peak[1] at
\mu