I am using scipy.stats.gaussian_kde to estimate a pdf for some data. The problem is that the resulting pdf takes values larger than 1. As far as I understand, this should not happen. Am I mistaken? If so why?
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asked Dec 29, 2010 at 20:27
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$\begingroup$ (+1 to the possible duplicate) Just to convey this quickly: Probability is defined as an area under a curve. A probability associated with the value of a PDF at a single point is multiplied by 0 (ie. the width of a line) so if anything the probability itself is 0. The linked thread gives excellent further elaboration on this. $\endgroup$– usεr11852Commented May 29, 2016 at 20:20
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You are mistaken. The CDF should not be greater than 1, but the PDF may be. Think, for example, of the PDF of a Gaussian random variable with mean zero and standard deviation $\sigma$: $$f(x) = \frac{1}{\sqrt{2\sigma\pi}}\exp(-\frac{x^2}{2\sigma^2})$$ if you make $\sigma$ very small, then for $x = 0$, the PDF is arbitrarily large!
answered Dec 29, 2010 at 20:33
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8$\begingroup$ Another possible source of confusion is that the pdf of a discrete random variable (also called pmf - probability mass function) indeed cannot exceed 1. $\endgroup$– AnikoCommented Dec 29, 2010 at 20:40
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$\begingroup$ @Aniko: This is indeed a source of confusion. I think I understand now. $\endgroup$ Commented Dec 29, 2010 at 20:48
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1$\begingroup$ This question is a duplicate of stats.stackexchange.com/q/4220/919 . $\endgroup$– whuber ♦Commented Dec 30, 2010 at 15:28