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For a continuous random variable with continuous PDF over the real axis and well defined CDF, are the mean, variance, and median always well defined?

Mean and variance do not always exist, e.g. for a Cauchy random variable. But what about the median?

Since the the median $m$ satisfies $$F_X(m)=\int\limits_{-\infty}^{m} f_X(x)dx=0.5,$$ can I say there must be a value $m$ satisfying the condition since it is continuous random variable with well defined CDF?

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Since you already apparently answered your own question in respect of everything but the median, I'll address that.

[However, while the mean for the Cauchy is undefined, I'd contend that it's possible to argue that the variance of the Cauchy is infinity (while its usually given as undefined because the mean is undefined, I think we could at least make an argument that it should be infinity, since variance can be defined without reference to the mean). I don't know that a value of infinity is necessarily poorly defined, but "undefined" would certainly seem to be. In any case, there will be situations where the variance doesn't exist, so either way your point is still okay.]

Note that the definition of the median is not unique. If there's a region with density zero, with half the probability either side of that region, the definition of the median you give is satisfied by every $m$ in the region.

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So the question arises as to whether that's enough to consider that definition of the median "not well-defined".

Wikipedia gives:

an expression is well-defined if it is unambiguous and its objects are independent of their representation

So the question comes down to whether we regard a definition we specify as "the median" is unambiguous when it can be any value in an interval. Perhaps "a median" would be more appropriate.

(Of course we can follow some convention and define it uniquely, but we're dealing with the definition in your question.)

Judging by some of the examples on that wikipedia page, I think it can be argued that the median isn't well-defined.

Note that my discussion is entirely based on taking it to be an attempt to define the median as a point. If we allow that the median is more generally an interval, as Henry suggested in comments below, then it's well-defined.

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    $\begingroup$ I would say that the second moment of a Cauchy random variable is infinite, but that its variance is not defined (since it depends on the undefined mean). $\endgroup$
    – Henry
    Commented Sep 6, 2014 at 12:41
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    $\begingroup$ As for the median, you could say that the median is a closed interval, and in many examples that interval reduces to a single point. $\endgroup$
    – Henry
    Commented Sep 6, 2014 at 12:43
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    $\begingroup$ @Henry The variance of any distribution $F$ can be defined without reference to the mean as $\lim_{a,b,c,d\to\infty}\mathbb{E}(I_{(-a,b)}(X)I_{(-c,d)}(Y)(X-Y)^2)/2$ for iid $X,Y\sim F$. For the Cauchy, this is the limit of $\frac{1}{2} \left(\frac{1}{2} \log\left(\frac{a^2+1}{b^2+1}\right) \log \left(\frac{d^2+1}{c^2+1}\right)+\left(\tan ^{-1}(a)+\tan ^{-1}(b)\right) \left(c-2 \tan ^{-1}(c)+d-2 \tan ^{-1}(d)\right)+(a+b)\left(\tan ^{-1}(c)+\tan ^{-1}(d)\right)\right)$. Since all such limits independently grow without bound, the Cauchy variance diverges ("is infinite"). $\endgroup$
    – whuber
    Commented Sep 6, 2014 at 18:35
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    $\begingroup$ Glen, this might seem like a pedantic nicety but it can resolve a potential point of confusion: although the median might not be well-defined as a number, it is--as @Henry points out--perfectly well-defined as an interval. Moreover, once some convention is adopted (such as using the midpoint of that interval), the median-qua-number becomes well-defined, too (albeit somewhat arbitrary). $\endgroup$
    – whuber
    Commented Sep 6, 2014 at 18:37
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    $\begingroup$ @whuber my treatment was entirely based on taking it to be an attempt to define the median as a point. If we allow that the median is more generally an interval, the objection in my post as, well, pointless. I've added some discussion. $\endgroup$
    – Glen_b
    Commented Sep 6, 2014 at 22:27

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