6

Commonly, the probability density function (PDF) is used when dealing with continuous random variables, while the probability mass function (PMF) is used for discrete random variables.

This is the reason they are called “density function” and “mass function” respectively.

However, my professor would talk about a “continuous PDF” and a “discrete PDF”, instead of a PDF and PMF.

It seems that my professor is not the only one to use the term “discrete probability density function”. It is also used in these UBC lecture notes.

Is it correct to call the probability mass function by the name “discrete probability density function”?

hb20007
  • 808
  • 4
  • 18
  • 34

1 Answers1

0

Let $X$ be a discrete random variable with probability mass function $p_X : \mathcal{X} \to [0,1]$, where $\mathcal{X}$ is a discrete set (possibly countably infinite). Random variable $X$ can be thought of as a continuous random variable with the following probability density function

$$f_X (x) = \sum_{x_k \in \mathcal{X}} p_X (x_k) \, \delta (x - x_k)$$

where $\delta$ is the Dirac delta. One could call $f_X$ a discrete probability density function, as its support is a discrete set. However, do note that $f_X$ is a generalized function, not a proper function.

Rodrigo de Azevedo
  • 23,223
  • But then this would not be a probability mass function. I was asking whether the simple probability mass function could also be called by the name 'discrete probability density function' – hb20007 Jun 22 '16 at 12:11
  • Sorry but I fail to see the advantages of equalling a function (your LHS) with a measure (your RHS), only to declare right away that the function is actually a generalized function, that is, an object for which we have no definition except the fact that it is not a function... :-) – Did Jun 22 '16 at 13:49
  • @Did If you want to compute the PDF of $Z = X + Y$, where $X,Y$ are independent, $X$ is discrete and $Y$ is continuous, then the PDF of $Z$ is the convolution of $f_X$ and $f_Y$, which is $$f_Z (z) = \sum_{x_k \in \mathcal{X}} p_X (x_k) , f_Y (z - x_k)$$ – Rodrigo de Azevedo Jun 22 '16 at 13:55
  • Yes, the PDF of Z is then the function which is the convolution of the distribution of X (which is not a function, but a measure) with the PDF of Y (which is a function). – Did Jun 22 '16 at 14:24
  • @Did Do you want me to replace "PDF" with "PDGF"? – Rodrigo de Azevedo Jun 22 '16 at 14:36
  • (Sighs.) $ $ $ $ – Did Jun 22 '16 at 17:03