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Objective: Just to help close a hole in my statistical knowledge (as opposed to having a practical subject matter need), I want to know what are some of the ways to display what I am calling a probability density function when the probability density function does not exist for one or more values of the random variable.

However, maybe the presumption that one can do this in the first place should be answered first. Yes, the cumulative distribution function can be plotted. And the derivative of that function for the example below exists for all points but one.

But if the derivative doesn't exist for a single point, does that mean the pdf doesn't exist for the random variable? If so, can I get around that by simply saying I have a "quasi-pdf" ? I find it hard to believe this isn't legitimate. But I've been wrong many times before. Maybe this is just a big hole in my knowledge of random variables.

Example:

Suppose $X \sim N (\mu, \sigma^2) $ and that $Y = \max (0, X) $. The CDF of $Y$ is

$$\begin{eqnarray*} \text{Pr}(Y \leq y) & = & 0 \quad \quad \quad \quad \quad \enspace \text{for } y<0 \notag \\ & = & \Phi\left((y-\mu)/\sigma\right) \text{ for } y\geq 0 \notag \\ \end{eqnarray*}$$

where $\Phi()$ is the standard normal cumulative distribution function.

What are reasonable ways to plot the "PDF" (the derivative of the CDF) when the PDF does not exist at $Y=0$? (A subquestion is "should" one attempt such a plot. I say absolutely "Yes" while at least one other says "No".)

Here is one way simply using a "dot" at the point (0,0) and a text explanation in the figure:

PDF of censored random variable

Are there standard/reasonable ways to do this which might include something (a circle or rectangle) proportional to the size of the probability mass?

This is related to a comment I made (and the associated discussion) on Mathematica StackExchange.

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    $\begingroup$ This article might be of interest $\endgroup$ – angryavian Oct 11 '20 at 5:34
  • $\begingroup$ @angryavian Excellent! Thanks. It seems from that article that adding explanations and a second vertical axis for the probability mass portion is a reasonable way to go. $\endgroup$ – JimB Oct 11 '20 at 5:38
  • $\begingroup$ The pdf does not exist wrt the Lebesgue measure since the rv has an atom at zero. $\endgroup$ – Xi'an Oct 11 '20 at 14:37
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    $\begingroup$ What is the purpose of the plot and who is the intended audience? This information is an important factor in evaluating the relative merits of potential solutions. $\endgroup$ – whuber Oct 11 '20 at 14:41
  • $\begingroup$ @whuber. Good point and I totally agree (and I ask that very question with all of the scientists that I deal with). In this case it's simply trying to fill in an apparent hole in my knowledge of random variables. I've now added an "Objective" section to my post. $\endgroup$ – JimB Oct 11 '20 at 17:43
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If your main objective is to understand what's going on with these mixed continuous-and-discrete "pdfs," then you might want to consider learning some measure theory. The article that @angryavian shared in the comments is probably a sufficient answer to your stated question about plotting these, but you might want to consider using the graphical representation of a Dirac delta (generalized) function instead (an arrow) instead of the line-point representation that the article uses. While strictly speaking, the probability density function (pdf) does not exist in your example, you can consider the "pdf" to be the generalized function (Schwarz distribution) given by $$ f(x) = \delta(x)\int_{-\infty}^0 \mathcal{N}(\mu, \sigma^2)(x) dx + I_{[0,\infty)}(x) \mathcal{N}(\mu, \sigma^2)(x) $$ where $\mathcal{N}(\mu, \sigma^2)$ is the normal distribution pdf, $I_{[0,\infty)}$ is the indicator function for the set $[0,\infty)$, and $\delta$ is the Dirac delta (generalized) function. This generalized function $f$ can be used as a "pdf" in the same way you would use a true pdf because $$ \int_A f(x)dx = P(A) \text{ for all measurable sets } A \subset \mathbb{R}. $$ And using the arrow representation of a Dirac delta, you can graph $f$ as:

enter image description here

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  • $\begingroup$ Thanks! That clears up the issue for me. (Although real clarity will be when I learn a bit more about measure theory.) $\endgroup$ – JimB Oct 12 '20 at 1:57

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