# Find the CDF of $Z=\{X_1+X_2, X_1<X_3\}$

Let $$X_1, X_2$$ and $$X_3$$ be three independent exponential random variables. The PDF and CDF of $$X_i$$ with parameter $$\beta_i$$ are $$f_{X_i}(x_i)=\beta_i e^{-\beta_i x_i}$$

$$F_{X_i}(x_i)=1-e^{-\beta_i x_i}$$

What is the CDF of random variable $$Z$$ define the sum of $$X_1$$ and $$X_2$$ where $$X_1$$ is less then $$X_3$$. $$Z=\{X_1+X_2, X_1.

Thanks.

• What are your thoughts? What have you tried? – Easymode44 Mar 14 at 13:19
• I want to find the probability that $X_1+X_2\leq z$ where $X_1$ is upper bound or less then another random variable $X_3$. – Monir Mar 14 at 13:21
• Yes, I can see that. Where exactly are you stuck? – Easymode44 Mar 14 at 13:22
• I tride $$\int_{x_3=0}^{\infty}\int_{x_2=0}^{x_3}\left(\int_{x_2=0}^{z-x_1}f_{X_2}(x_2) dx_2\right)f_{X_1}(x_1) dx_1f_{X_3}(x_3) dx_2.$$ – Monir Mar 14 at 13:24
• I stuck in the last integral. And i think we need to take all possible cases – Monir Mar 14 at 13:25