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I have this random variable ##\beta=\sum_{k=1}^K\alpha_k##, where ##\{\alpha_k\}_{k=1}^{K}## are i.i.d. random variables with CDF ##F_{\alpha}(\alpha)=1-\frac{1}{\alpha+1}## and PDF ##\frac{1}{(1+\alpha)^2}##. I want to find the CDF of the random variable ##\beta##. So, I used the Moment Generating Function (MGF) which is defined as:

[tex]\mathcal{M}_{\beta}(s)=E_{\beta}[e^{s\beta}]=\prod_{k=1}^K\mathcal{M}_{\alpha_k}(s)[/tex]

The CDF then can be found using the inverse Laplace transform of ##\frac{\mathcal{M}_{\beta}(-s)}{s}##. After doing all of this I ended up with

[tex]\frac{\mathcal{M}_{\beta}(-s)}{s}=\sum_{l=0}^K{K\choose l}s^{l-1}\,e^{ls}\text{Ei}^l(-s)[/tex]

where ##\text{Ei}(x)## is the exponential integral function. I have two questions:

- Is there another more simple way of computing the CDF?
- Can I evaluate the inverse Laplace transform of ##\frac{\mathcal{M}_{\beta}(-s)}{s}## other than numerically?