I was doing my best to calculate moments of phase-type distribution. Density of phase-type distribution is $$f(x)=\alpha e^{Sx}S_{0}$$ ($\alpha$ is $1\times m$ vector; $S_{0}$ is $m\times 1$ vector; $S$ is $m\times m$ matrix) for all $x>0$, where $e^{Sx}$ is matrix expotential. So first of all I checked if integral of density over $\mathbb{R}$ is equal to $1$: $$\int_{0}^{\infty}\alpha e^{Sx}S_{0} \ dx = [\alpha S^{-1}e^{Sx}S_{0}]_{x=0}^{\infty} = -\alpha S^{-1}S_{0}$$ So, the following equality must hold: $-\alpha S^{-1}S_{0}=1$ Now I tried to compute expected value: $$\int_{0}^{\infty}x\alpha e^{Sx}S_{0} \ dx = [x\alpha S^{-1}e^{Sx}S_{0}]_{x=0}^{\infty} - \int_{0}^{\infty}\alpha S^{-1}e^{Sx}S_{0} \ dx = [-\alpha S^{-2}e^{Sx}S_{0}]_{x=0}^{\infty} = \alpha S^{-2}S_{0}$$ But according to wikipedia it should be equal to: $$-\alpha S^{-1}\textbf{1}$$
Can anyone tell me where is the mistake?
