Number of athletes concluding a marathon after 'x' minutes

hugomcp74@gmail.com

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Suppose that there is a set of N athletes that will run a marathon. Depending of its skills, the time taken by each athlete to complete the marathon is modeled by a Gaussian distribution (mu_i, sigma_i). There will be N Gaussian distributions.

My doubt is about an analytic way to answer to: "What's the probability that after 'x' minutes, K out of N athletes having concluded the marathon?"

The unique way I can do this is by enumerating all the possible combinations of outcomes, but clearly cannot be done for large numbers.

P(Athlete 1 finished, Athlete 2 finished,...Athlete N finished)
P(Athlete 1 Not finished, Athlete 2 finished,...Athlete N finished)
...

When simulating this problem, I get a 3D sigmoid that makes sense, but I was looking for the formula (that might mixture in some way the Gaussians parameters).Screen Shot 2018-09-16 at 19.54.29.jpg
 
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