Kernel density estimate takes values larger than 1 [duplicate]

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?

• (+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. May 29, 2016 at 20:20

1 Answer

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!

• Another possible source of confusion is that the pdf of a discrete random variable (also called pmf - probability mass function) indeed cannot exceed 1. Dec 29, 2010 at 20:40
• @Aniko: This is indeed a source of confusion. I think I understand now. Dec 29, 2010 at 20:48
• This question is a duplicate of stats.stackexchange.com/q/4220/919 .
– whuber
Dec 30, 2010 at 15:28