Given three independent, non-negative (continuous) random variables $v_1$, $v_2$ and $v_3$ with PDFs $f_{v_1}$, $f_{v_2}$ and $f_{v_3}$ respectively, I want to calculate the probability of satisfying a system of linear inequalities.

For example, what is the probability $\mathbb{P}(v_1 \leq q_1 \land [ v_1 + v_3 \leq q_2 \lor v_2 + v_4 \leq q_2])$ of satisfying $v_1 \leq q_1 \;\textbf{and}\; [ v_1 + v_3 \leq q_2 \;\textbf{or}\; v_2 + v_3 \leq q_2 ]$ where $q_i \in \mathbb{Q}$ are rational constants. (Sorry for my abuse of notation!)

I can easily calculate the probability for each of the three inequalities (using convolutions) by $\mathbb{P}(v_1 \leq q_1) = \int_{-\infty}^{q_1} f_{v_1}(t)\; dt$,

$\mathbb{P}(v_1 + v_3 \leq q_2) = \int_{-\infty}^{q_2} f_{v_1}(t) (\int_{-\infty}^{q_2 - t} f_{v_3}(u)\; du)\; dt$, and

$\mathbb{P}(v_2 + v_3 \leq q_2) = \int_{-\infty}^{q_2} f_{v_2}(t) ( \int_{-\infty}^{q_2 - t} f_{v_2}(u)\; du)\; dt$

Let $A$ be the event that $v_1 + v_3 \leq q_2$ and $B$ the event that $v_2 + v_3 \leq q_2$.

The sum rule tells us that $\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)$, and we know that $\mathbb{P}(A \cap B) = \mathbb{P}(A|B) \mathbb{P}(B) = \mathbb{P}(B | A) \mathbb{P}(A)$.

Is it possible to calculate the conditional probabilities $\mathbb{P}(A|B)$ and $\mathbb{P}(B|A)$ knowing only the marginal distributions for $v_1$, $v_2$ and $v_3$?


You did know how to calculate the conditional distribution by your statement : $$\mathbb{P}(v_1+v_3 \le q_2) = \int_{-\infty}^{q_2}f_1(t) \int_{-\infty}^{q_2-t}f_3(u)dudt $$ It is indeed $$\mathbb{P}(v_1+v_3 \le q_2) = \mathbb{E}[\mathbb{E}(1_{\{v_1+v_3 \le q_2\}}|v_1)]$$ Same calculation for the set $A\cap B$ $$\mathbb{P}(v_1+v_3 \le q_2 \cap v_2+v_3 \le q_2 ) = \mathbb{E}[\mathbb{E}(1_{\{v_1+v_3 \le q_2 \cap v_2+v_3 \le q_2\}}|v_3)]$$ In general, since all variable are independent, so the joint density is the multiplication of all densities. If the event is not too complicated, one can just calculate the integral by 'freezing' variable and calculate the integration on a single variable at a time.


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