## Roots of Two and the Median of the Minimum

Yet another weird math observation. As *k* becomes large (like *k*=1000 has less than 0.034% relative error):

I haven’t actually looked into why this is, but it means that:

So why am I looking at this? Well, I’m really looking at the finding the median of the minimum of *k* independent, identically-distributed random varaibles. With *k* such variables:

The question is, what value *a* is the median of the minimum? Let’s define and solve for it:

If it’s the minimum of many independent variables, it’s then . So, as long as we can find the value *a* at percentile *p* of the original distribution, we’ve found the median of this distribution of the minimum. Weird stuff.