Definition talk:Probability Density Function

Is this defined only when that complicated limit actually exists? Is it the case that it does always exist? --prime mover (talk) 13:55, 20 December 2012 (UTC)


 * Corrected to point out that it doesn't always exist. The definition is neater when the pdf is considered as the derivative of the cdf, but I was going to add that as a theorem, with a sufficient condition for the pdf existing is that the cdf is differentiable. If you have a more elegant approach by all means please suggest it. The motivation of this definition is to get around the problem that $Pr(X=c) = 0$, should I mention it on the page so that the limit isn't coming out of nowhere? --GFauxPas (talk) 14:37, 20 December 2012 (UTC)


 * Having thought about it, if $X$ is continuous, then it probably does exist througout. No worries, just thought I'd ask. If you're quoting what Bean says, then trust him not me ... --prime mover (talk) 17:37, 20 December 2012 (UTC)


 * It wasn't just what you said; I realize now I was confusing piecewise-continuous and continuous. I'll fix it later --GFauxPas (talk) 17:48, 20 December 2012 (UTC)


 * Okay, I have enough clarity to fix the page. Now, what's the best way to present the following definition? I'm not sure how to present it clearly and rigorously:

Let $X$ be piecewise continuous.


 * $\forall x \in \R: f_X \left({x}\right) = \begin{cases}

\ds \lim_{\epsilon \to 0^+} \frac{\Pr \left({x-\frac \epsilon 2 \le X \le x + \frac \epsilon 2}\right)} \epsilon & : \text{all $x \in \Omega_X$ except for countably many $x$ at which point $f_X$ has a removable discontinuity} \\ \text {doesn't matter, as long as it's a real number between 0 and 1} &: \text{the countably many $x \in \Omega_X$ that I mentioned in the above line} \\ 0 & : x \notin \Omega_X \end{cases}$

--GFauxPas (talk) 14:18, 21 December 2012 (UTC)


 * So you are defining $f_X$, and in that definition, you refer to it having a discontinuity? That's not a very good idea. Not sure how to circumvent that, though. --Lord_Farin (talk) 15:39, 21 December 2012 (UTC)
 * Why is it a bad idea? It's just saying that $X$ need not be everywhere continuous to be a probability measure --GFauxPas (talk) 23:02, 22 December 2012 (UTC)

This seems to basically define (at least A.E. - a PDF is not unique) the PDF as the derivative of the CDF, or at least some one-sided derivative. (rewrite numerator as $\map {F_X} {x + \epsilon/2} - \map {F_X} {x - \epsilon/2}$) I'd prefer this to be a theorem, but I suppose it works as a definition too. There's machinery missing for this though: first we are talking about absolutely continuous random variables, not general continuous random variables, which implies that the CDF is absolutely continuous. (I might add this as a second definition, but definitely at least as a theorem) A result (not yet on this site) in real analysis then says that the CDF is almost everywhere differentiable, meaning that you can define $f_X$ on the complement of a null set (which may be countable or finite [or indeed empty] but could well be bigger than that) by the derivative of the CDF. You can then fill in the missing points as discussed above. The actual value of $f_X$ on this null set is arbitrary, (integration doesn't care about functions that differ only on a null set) so it'd be conventional to send these points to $0$. Caliburn (talk) 16:17, 29 December 2021 (UTC)


 * I think I had a lapse of concentration, I meant to finish this off with: the proper way to define a PDF seems to be as a particular Radon-Nikodym derivative of the probability distribution of $X$ with respect to the Lebesgue measure of $\R$. I'm still yet to properly set up Radon-Nikodym derivatives, but the proof of their existence and essential uniqueness is already up at Radon-Nikodym Theorem. I'll decide what to do with what's already here closer to the time. Caliburn (talk) 16:31, 29 December 2021 (UTC)