I saw that one of the properties of probability density function is that it is always positive.

But I am not sure how to prove that?

  • 5
    $\begingroup$ Small modification: it is non-negative, not necessarily positive. $\endgroup$ Commented Sep 29, 2013 at 15:13

2 Answers 2


By definition the probability density function is the derivative of the distribution function. But distribution function is an increasing function on $\mathbb{R}$ thus its derivative is always positive.

  • 8
    $\begingroup$ Not quite. Distribution functions are nondecreasing functions, not increasing functions, and their derivatives are always nonnegative, not always positive. See the comment by Andre on the main question. $\endgroup$ Commented Sep 29, 2013 at 15:24
  • $\begingroup$ yes, of cource, thanks for the refinement $\endgroup$
    – Leox
    Commented Sep 29, 2013 at 16:16
  • $\begingroup$ very great interpretation $\endgroup$
    – Johnny Ji
    Commented Dec 14, 2017 at 9:38
  • 1
    $\begingroup$ a pdf is not necessary the derivative of the cdf at all points. See chapter 1 in "Statistical Inference" $\endgroup$
    – Jesse
    Commented Nov 8, 2020 at 5:00

Assume that probability density of X is -ve in the interval (a, b). Now, let us define an event A such X lies between a and b. This is nothing but integral of density between a and b. Thus, P(a < X < b) < 0. But by definition, probability can never be negative. Thus, density can never be negative.


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