Let $M \in \mathcal { M } _ { n } ( \mathbb { R } )$ and $N \in \mathcal { M } _ { n } ( \mathbb { R } )$ be two stochastic matrices, $X \in \mathbb { R } ^ { n }$ a probability distribution and $\alpha \in [ 0,1 ]$. Show that $X M$ is a probability distribution.