The weighted Lindley distribution has probability density function
$$
f(z;\theta)=\frac{\theta}{2\Gamma(\theta)}a_{\theta}^{-b_{\theta}-1}z^{b_{\theta}-1}(1+z)\exp\left(-\frac{z}{a_{\theta}}\right), \quad z, \theta>0,
$$
where \(a_{\theta}=\frac{\theta(\theta+4)}{2(\theta+2)}\) and \(b_{\theta}=\frac{4}{\theta(\theta+4)}\). Under
this parametrization, E(Z)=1 and Var(Z)=\(\theta\).