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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 $X_3$. $$Z=\{X_1+X_2, X_1<X_3\}$$.

Thanks.

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

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