# Total Probability Theorem over multiple variables

Let $$X$$ and $$Z$$ be continuous random variables. Let a conditional PDF be defined as $$f\left(X\big|Z\right)$$. To use total probability theorem (TBT), we introduce a discrete random variable $$\Theta$$ as given $$f\left(X\big|Z\right) = \sum_{\theta}f\left(X,\Theta = \theta\big|Z\right)$$ The question is, how to use TBT of we have two or more discrete random variables $$\Theta_1$$, $$\Theta_2$$, $$\ldots$$, $$\Theta_N$$?

• $$f\left(X\big|Z\right) = \sum_{\theta_1} \sum_{\theta_2} \ldots \sum_{\theta_N} f\left( X, \Theta_1 = \theta_1, \Theta_2 = \theta_2, \ldots , \Theta_N = \theta_N \big| Z \right)$$ – Ertxiem May 22 at 11:54