Probability of multiple distributions

Question

I have a question concerning the calculation of the probability of an event among multiple probability distributions. Let's say we have two continuous probability distributions $D_1$ and $D_2$, given by the functions $f_1$ and $f_2$, respectively, and their two random variables $X_1$ and $X_2$, belonging to $D_1$ and $D_2$, respectively. I would like to know how to calculate the probability of $x_1 + x_2 > T$, i.e., $P(x_1 + x_2 > T)$.

Thanks for your attention, case my notations are not understandable or any point is not clear, please let me know.

ps.: I am not a mathematician.

Answer 1

Plot the product of the two (independent) distributions as a function of $X_1$ and $X_2$. Then consider the criterion line (plane) $X_1 + X_2 = T$.

The probability $P(X_1 + X_2 \geq T)$ is thus the integral of the joint probability above that plane (to the upper right) divided by the integral over the whole space (i.e., the normalization).

$$P(X_1 + X_2 \geq T ) = \frac{\int\limits_{x=0}^\infty \int\limits_{y = \max[0,T-x]}^\infty D_1(x) D_2(y)\ dx\ dy}{\int\limits_{x=0}^\infty \int\limits_{y = 0}^\infty D_1(x) D_2(y)\ dx\ dy}$$

Clear?