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Is it incorrect to say that a Poisson (or any discrete) distribution has a probability density function? I thought that a discrete distribution has a PMF and not a PDF.

Sycorax
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Joe Exotic
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1 Answers1

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No, you can't call it as density, because $p_X(x)$ is not a density value for discrete distributions, it's probability. The term probability mass function is not used extensively in the literature, but that shouldn't confuse and make you to refer it as density function.

But, there is a notion called generalized density which uses Dirac delta functions to make discrete distributions look like continuous.

Nick Cox
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gunes
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  • So just to be clear, you're saying "No, it's not correct", right? – Joe Exotic Apr 15 '20 at 12:57
  • Yes, I'm saying: 'No, it's not correct' – gunes Apr 15 '20 at 12:58
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    Indeed, or on the contrary, you **can** call it density. It's density too, but with respect to counting measure. SInce Parzen, some people have preferred _probability mass function_, and that's a fine term too, especially if it is helpful, but density is a flexible concept regardless. (Think density in physical science, population density in demography and ecology, drainage density in hydrology and geomorphology, to say nothing of a variety of flavours in mathematics.) Many probability books use the broad sense, e.g. Peter Whittle's intermediate text. – Nick Cox Apr 15 '20 at 13:05
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    Good point, I see different usages as well, but still with its CDF being not absolutely continuous, Poisson distribution doesn't fit very well in density notion in rigorous sense I believe. – gunes Apr 15 '20 at 13:15
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    Rigorous treatment lies in the hands of authors; and is not inherent in particular terminology. It's my impression that the more measure theory you know, the more likely you are to follow a general idea of density (my own knowledge is almost of measure 0). – Nick Cox Apr 15 '20 at 14:34