Consider independent random variables $X_1, \ldots, X_n$, each of them exponentially distributed with parameters $\lambda_1, \ldots, \lambda_n$. Let us consider the sum $Y= X_1 + \ldots + X_n$.
(1) In the case where all the rates are equal ($\lambda_i = \lambda$ for all $i$), we know that the distribution is a Gamma distribution:
$$
f_Y(t) = \lambda e^{-\lambda t} \frac{(\lambda t)^{n-1}}{(n-1)!}.
$$
(2) In the case where all the rates are different $(\lambda_i \neq \lambda_j $ for all $i,j$), we can also calculate the distribution of $Y$. This time it is given by the hypoexponential distribution:
$$
f_Y(t) = \sum_{i=1}^{n} \left( \prod_{j\neq i} \frac{\lambda_j}{\lambda_j – \lambda_i} \right) \lambda_i e^{-\lambda_i t},
$$
where the product is over all $1 \leq j \leq n$, $j \neq i$. For an induction proof see Ross.
(3) Now consider the case in which some rates are equal to others but they are not all equal, i.e. the number of unique rates is between $2$ and $n-1$. Is there a general formula for the distribution of the sum in this case?
I tried approaching the problem by computing convolutions and trying to find a pattern in the distributions. For example, for two variables $X_1, X_2$ I would calculate $f_{X_1+X_2}(t) = \int\limits_{0}^{t} f_{X_1}(s) f_{X_2}(t-s) ds$, and then do this iteratively for more and more variables where some but not all rates are similar. From the results I obtained I came up with the following conjecture:
Assume that there exist $i,j$ such that $\lambda_i = \lambda_j$ and that there exist $m,n,$ such that $\lambda_m \neq \lambda_n$. Define $m$ as the number of distinct rates, $2 \leq m \leq n-1$. Define $n_j$ as the number of variables with rate $\lambda_j$, $1 \leq n_j \leq n$ and $1 \leq j \leq m$. Hence the system is specified by $(m, n_1, \ldots, n_m)$ and rates $(\lambda_1, \ldots, \lambda_m)$. Then the distribution of $Y$ is given by
$$
f_Y(t) = \sum_{i=1}^{m} \left[ \prod_{j\neq i} \left(\frac{\lambda_j}{\lambda_j – \lambda_i}\right)^{n_j} \right] \lambda_i^{n_i} \left[ \sum\limits_{k=1}^{n_i} \frac{t^{k-1}}{(k-1)!} \right] e^{-\lambda_i t},
$$
I was not able to proof this by induction or verify it in some other way than by checking results for a small number of variables. Could you prove or disprove the conjecture I made above?
Linked question
Distribution of sum of exponential variables with different parameters: a similar question, but it only considers a special case with parameters $(\lambda, \lambda/2, \lambda/3, \ldots, \lambda/n)$, which belongs to case (2) above. I would like to know the general case for arbitrary $\lambda_i$'s.
Best Answer
The general formula you are looking for is on equation (1.1) on the paper of Smaili et. al (2016). On that paper they review examples of your 3 cases plus one more intermediate case.