I have this random variable
$$Y = \sum_{k=1}^KX_k$$
where $X_k$ are i.i.d. random variables with CDF and PDF $F_X(x)$ and $f_X(x)$, respectively.
In my application, the CDF of $Y$ denoted by $F_Y(y)=\text{Pr}\left[Y\leq y\right] = \text{Pr}\left[\sum_{k=1}^KX_k\leq y\right]$ is not easy to evaluate directly. So, I am trying to find a tight lower and upper bound, such that the exact CDF is very close to both.
One obvious options for the lower and upper bounds are
$$\text{Pr}\left[K\min_k X_k\leq y\right]\geq \text{Pr}\left[\sum_{k=1}^KX_k\leq y\right] \geq \text{Pr}\left[K\max_k X_k\leq y\right]$$
I have done some simulations on the lower bound, and it is lose. What are other lower and upper bounds that I can use?
EDIT: In my application, $X_k=A_k/B_k$, where $A_k$ and $B_k$ are i.i.d exponential random variables with parameter 1, and the means of such random variables don't exist.