DES - Demonstration of the Exponential Distribution of the Superposition

Arguments:

DES - Exponential Distribution

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Given the event $X_i\sim Exp\left(\frac{1}{{\lambda }_i}\right)$ for $i=1,2,3,\dots ,n$ What is the distribution of $X$ ?

Now, remember that being an exponential distribution we have that:

$$P(X_i>t)=-e^{-{\lambda }_{i}t}$$

So:

$$=1-e^{-{\lambda }_{1}t}\cdot -e^{-{\lambda }_{2}t}\cdot \dots \cdot -e^{-{\lambda }_{n}t}$$

Which can be seen as:

$$=1-e^{-\left({\lambda }_1+{\lambda }_2+...+{\lambda }_n\right)t}$$

So the distribution of the superposition is:

$$P(X\le t)=1-e^{-\left({\lambda }_1+{\lambda }_2+...+{\lambda }_n\right)t}$$

$\Rightarrow$ The superposition of $X_1$, $X_2$, …, $X_n$ has an exponential distribution with rate:

$$\Lambda ={\lambda }_1+{\lambda }_2+...+{\lambda }_n$$