What is the maximum value a probability density function and a probability mass function can have? Also, if we take a pdf of a continuous distribution, will the pdf always be continuous? I am a little confused between the physical nature of these functions. I would really appreciate if someone could give an intuitive explanation for these, and how they relate to each other.

  • $\begingroup$ Since probability can't be greater than 1 (an event can have at most a 100% chance of happening), the maximum value either of those functions could possibly take is 1. $\endgroup$ – Dave Oct 22 '16 at 12:05
  • $\begingroup$ @Dave for pmf yes, for pdf no. $\endgroup$ – user251257 Oct 22 '16 at 12:59
  • $\begingroup$ Pdf is non-negative, measurable, and integrates to $1$. Nothing more, nothing less. $\endgroup$ – user251257 Oct 22 '16 at 13:03
  • $\begingroup$ @user251257 what if the pdf was equal to 1 between 0 and 1, and 0 everywhere else? $\endgroup$ – Dave Oct 22 '16 at 13:04
  • $\begingroup$ @Dave the maximum of a pdf could be any positive number. I don't get your comment. $\endgroup$ – user251257 Oct 22 '16 at 13:05

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