TITLE: Why is probability density function is always positive?
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: 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.