4

Given a random variable X, which (for the sake of simplicity) we'll say has some continuous distribution (whose pdf is f(x)), is it possible to express the median in terms of the distrbution's moments? Or is there a result/theorem that shows there is no elementary closed form expression (in general)?

Thanks!

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
Matt Brenneman
  • 472
  • 3
  • 12
  • 3
    There exist pairs of random variables $X, X^\prime$ having the same moments *of every positive integral order* yet which have different medians. See, for instance, the example at https://stats.stackexchange.com/a/25017/919 and consider the cases $\lambda=0$ (median is near 0.55) and $\lambda=1$ (median is near 0.63). – whuber Nov 08 '20 at 17:21
  • 1
    @whuber Thank you for that example! – Matt Brenneman Nov 08 '20 at 19:56

1 Answers1

3

Why do you expect there to be a connection? Say (without loss of generality) the only thing you know about the distribution of a random variable $X$ is that its median is zero. To simplify lets assume a continuois distribution with cdf (cumulative distribution function $$\DeclareMathOperator{\P}{\mathbb{P}} F(x)= \P(X \le x)=\frac12 $$ This restrains the distribution very little, and (assuming expectation of $X$ exist) it is now an exercise for you to find $F$ satisfying above with expectation $-1, 0, 1,2,3,4, \dotsc $ and many other values.

When you have constructed those examples, you will have answered your own question!

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
  • I appreciate your answer, but maybe you missed the point. Obviously, to compute the median I dont know just the value of the median (I also know the distribution itself). Since the median depends on the distribution, it is not obvious, a prori, to me that it would not necessarily related to the moments (which are just terms in the Taylor series expansion of the pdf, upon which the median depends). – Matt Brenneman Nov 08 '20 at 17:09
  • Do you mean you know **all** the moments? If so, please make that clear with an edit! – kjetil b halvorsen Nov 08 '20 at 17:32