Distribution of sum of exponential variables with different parameters

Question

We have $k$ independent random variables with exponential distribution ($T_1, T_2, \ldots , T_k$), parameters of random variables are ($\lambda,\frac{\lambda}{2},\frac{\lambda}{3},\ldots,\frac{\lambda}{k}$), what is the distribution of new variable $T = T_1 + T_2 + \cdots + T_k $

Answer 1

you can use the main result here applied to the sequence $\lambda, \lambda/2,...,\lambda/k$, which for the general case states that

$$ h_{X_1,.,X_n}(x) = \left[ \prod_{i=1}^n \lambda_{i} \right] \sum_{j=\ 1}^n \frac{e^{-\lambda_i x}}{\prod_{k \neq j}^n(\lambda_i-\lambda_j)} $$

Answer 2

Hint: You could use convolution to calculate the distribution of two independent variables:

Assume $X$ follows $f(x)$, $Y$ follows $g(y)$, then $Z=X+Y$ follows

$$f_Z(z)=\int _{-\infty}^{+\infty}f_X(z-y)f_Y(y) \, dy.$$

Following this logic, you just do a serial integrations, then you would get the result.