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):

1-\frac{1}{\sqrt[k]{2}}\approx\frac{\ln 2}{k}

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

\sqrt[k]{2}\approx\frac{k}{k-\ln 2}

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:

P(\min(X_1,...,X_k)<a) = 1-P(X_1\ge a \wedge X_2\ge a \wedge ... \wedge X_k\ge a)
= 1-P(X_1\ge a)^k
= 1-(1-P(X_1 < a))^k

The question is, what value a is the median of the minimum?  Let’s define p=P(X_1 < a) and solve for it:

\frac{1}{2}=1-(1-p)^k
(1-p)^k=\frac{1}{2}
p = 1-\frac{1}{\sqrt[k]{2}}

If it’s the minimum of many independent variables, it’s then \approx\frac{\ln 2}{k}.  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.

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~ by Neil Dickson on August 31, 2009.

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