Given the following graph:
$$\lower{4ex}X\lower{2ex}\swarrow Y\lower{2ex}\searrow\lower{4ex}Z$$
define the probabilities such that Z and X are independent while the value of Y is unknown and prove that Z and X are indeed independent.
I know that if X and Z are independent then $P(X,Z)=P(X)\cdot P(Z)$. Also a node is conditionally independent of its non-descendants given its parents. But I'm not sure how to proceed with this.
Yes, indeed, the outline is :
$$\begin{align}\mathsf P(X,Z) & =\sum_Y \mathsf P(X,Y,Z) \\[0ex]&~~\vdots \\[1ex]&=\sum_Y \mathsf P(X)\,\mathsf P(Y\mid X,Z)\,\mathsf P(Z) \\[1ex]&= \mathsf P(X)\,\mathsf P(Z)\sum_Y\mathsf P(Y\mid X,Z) \\[1ex]&= \mathsf P(X)\,\mathsf P(Z) \end{align}$$
You just need to justify that step.