expectation of normal-wishart distribution

Question

I want to compute $ E[\mu\Lambda] $ for a normal-wishart distribution how can i compute it? A normal-wishart distribution is defined as below: $$ (\mu,\Lambda)=NW(\mu,\Lambda|\mu_0,\lambda,W,v)=N(\mu|\mu_0,(\lambda \Lambda)^{-1})W(\Lambda|W,v) $$ in which $N(.)$ means normal distribution and $W(.)$ means wishart distribution. Please help me. thanks

Answer 1

$$ E[\mu\Lambda] = E\bigl[E[\mu\Lambda \mid \Lambda]\bigr] = E\bigl[\Lambda E[\mu \mid \Lambda]\bigr] $$ One has $(\mu \mid \Lambda) \sim \mathcal{N}\bigl(\mu_0, (\lambda\Lambda)^{-1}\bigr)$, hence $E[\mu \mid \Lambda]=\mu_0$.

Hence $E[\mu\Lambda] = \mu_0 E[\Lambda] = \mu_0 v W$.