Let X$_1$ and X$_2$ be identically independent distributions(i.i.d) random variables with

$$\Bbb P(X_i \le x) = 1-x^{-1/2}, \quad x \ge 1 \ \text{and} \ i = 1,2 $$

Find $\Bbb P(X_1 + X_2 \le x)$.

I tried to finding the convolution of $f_{X_1}$ and $f_{X_2}$ (the density functions for $X_1$ and $X_2$) and integrating to get the CDF of $X_1$ + $X_2$ which is what I interpreted the question is asking for. When I integrate it, it got really messy and I could not finish the integration. Am I doing something wrong?

  • $\begingroup$ Wolframalpha gives the answer using hypergeometric functions. I think there is no hope to get simple answer. $\endgroup$ – NCh Feb 15 at 6:00
  • $\begingroup$ I actually just noticed that the exponent of -1/3 is incorrect. It’s suppose to be -1/2 if that makes a difference. I’ll make the correction now $\endgroup$ – Niko L Feb 15 at 13:15
  • $\begingroup$ wolframalpha.com/input/… $\endgroup$ – NCh Feb 15 at 14:36

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