1. ## Hyper-exponential Random Variables

Lets say we want to compute the probability density function for a hyper-exponential random variable for $n = 2$.

So $f_{X_{1} + X_{2}}(t) = \int_{0}^{t} f_{X_{1}}(s) f_{X_{2}}(t-s) \ ds$.

Where did the $t-s$ come from?

2. Originally Posted by shilz222
Lets say we want to compute the probability density function for a hyper-exponential random variable for $n = 2$.

So $f_{X_{1} + X_{2}}(t) = \int_{0}^{t} f_{X_{1}}(s) f_{X_{2}}(t-s) \ ds$.

Where did the $t-s$ come from?
By definition, to get $f_{X_{1} + X_{2}}(t)$ you take the convolution of $f_{X_1}$ and $f_{X_2}$.