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.

~ by Neil Dickson on August 31, 2009.

Posted in Math

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