Assume that $X_1$ and $X_2$ are independent random variables with given distribution $f(.)$ (say Normal distribution with $\mu_i$ and $\sigma_i$). I am stuck with the calculation of:
$P(\{X_1 \leq a\} \; \cap\; \{X_2 \leq b\} \; \cap\; \{c \leq X_1 + X_2 \leq a+b\} )$
where $c < a+b$. For brevity, let's say $Z_1 = \{X_1 \leq a\}$, $Z_2 = \{X_2 \leq b\}$ and $Z_3 = \{c \leq X_1 + X_2 \leq a+b\}$.
Using the product rule I can write:
$P(Z_1 \cap Z_2 \cap Z_3) = P(Z_1)\cdot P(Z_2|Z_1) \cdot P(Z_3|Z_1\cap Z_2)$
and since $X_1$ and $X_2$ are independent, then $P(Z_2|Z_1) = P(Z_2) $ and $P(Z_1 \cap Z_2) = P(Z_1)\cdot P(Z_2)$. But now I don't know how to calculate the third factor $P(Z_3|Z_1\cap Z_2)$ since, as far as I understand, $Z_3$ is not independent of $Z_1$ and $Z_2$. Is there any other way to solve this problem?
Thanks.