Curious what the cardinality of the set of pdfs is. Does it change depending on whether they have infinite/finite support?
Edit: Sorry for the poor wording, I mean given a fixed support, what is the cardinality of the set of pdfs on that support?
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I believe this is uncountable regardless. Take the uniform distribution with parameter $\theta$ defined by $U(0,\theta)$ where $\theta \in [0,1]$. Since $[0,1]$ is uncountable, so should the PDF, no? – Gregory Mar 10 '21 at 21:42
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I imagine the situation is "worse" with infinite support. – Gregory Mar 10 '21 at 21:43
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Sorry, I should have probably been more clear - I meant more for a fixed support is it uncountable? – Keith Wynroe Mar 10 '21 at 21:44
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Same reasoning... take the normal distribution with fixed variance and mean ranging between 0 and 1. The interval is uncountable so the number of PDFs is uncountable. – Gregory Mar 10 '21 at 21:45
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1@Gregory The more interesting question is, is the number of PDFs more than the number of reals? (The answer is yes, so the number of PDFs is more than just uncountable) – Rushabh Mehta Mar 10 '21 at 21:47
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Oh wow duh yeah. Dunno what I was thinking, thanks! – Keith Wynroe Mar 10 '21 at 21:47
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@DonThousand yes, you are right of course. – Gregory Mar 10 '21 at 21:52
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@ Don Thousand will you please provide a link to that fact. Thanks. – Suman Chakraborty Mar 10 '21 at 21:53
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1I don't think DonThousand's claim is true. CDFs are right-continuous and therefore determined by their values on rational points (right?), so the set of all real-valued random variables has cardinaity $|\Bbb R^\Bbb Q|=|\Bbb R|$. – Karl Mar 10 '21 at 22:02
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See this answer. – Karl Mar 10 '21 at 22:07
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@Karl Many PDFs have the same CDF... – Rushabh Mehta Mar 10 '21 at 22:24
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Oh! How is that? Doesn't the CDF uniquely determine the distribution? – Karl Mar 10 '21 at 22:29