# Notation for probability density

On one of my other questions here, I was criticized (and rightly so, as it was the source of my mistake) for using this notation for a continuous random variable $X$ with pdf $f(x)$: $$\mathbb{P}\{X\in dx\} = f(x) \mathop{dx} .$$ I was taught this notation by my teacher. The purpose of this notation is to stress the fact that technically, $\mathbb{P}\{X=x\}=0$ if $X$ is continuous. I believe the notation is shorthand for $\mathbb{P}\{X\in [x, x+dx)\} = f(x) \mathop{dx}$.

As I am only familiar with this notation and didn't know that other people did not use it, I would really appreciate it if someone could give me a better sense of what notation is more accepted/used. Is the $f$ and $F$ notation of pdf and cdf respectively widely used?

Given a pdf $f(x)$ for a continuous random variable $X$, its cdf is $F(x)=\int_{-\infty}^x f(t)dt=\mathbb{P}\{X<x\}$ (thanks for catching the typo, André!), which corresponds to the discrete case where $F(x) = \sum_{k=0}^x f(x) = \mathbb{P}\{X\le x\}$. Is there just no continuous analog for the discrete expression of $\mathbb{P}\{X=x\}=f(x)$? One piece of advice that I was given by another user was to deal primarily with cdfs and not with pdfs.

Thank you in advance, and apologies if this was not the right place to ask this question.

• You will be dealing with density functions a lot, there is no reason to avoid them. Just avoid thinking in terms of $\Pr(X=x)$. The remark about interval from $x$ to $x+dx$ is informally fine. And in answer to your question, yes, the notations $f(x)$, $F(x)$, or $f_X(x)$, $F_X(x)$ are standard for density, cdf. By the way, $F_X(x)=\int_{-\infty}^x f_X(t)\,dt$. Commented Jan 8, 2013 at 22:17
• It doesn't make any difference for continuous random variables, but for consistency of notation, I would suggest that you always use $P\{X \leq x\}$ as the value of $F_X(x)$, and so write $$F_X(x) = \int_{-\infty}^x f_X(t)\,\mathrm dt = P\{X\leq x\}$$ instead of $$F_X(x) = \int_{-\infty}^x f_X(t)\,\mathrm dt = P\{X< x\}$$ as you have it. Commented Jan 9, 2013 at 0:02
• Also, $P\{X \in [x, x+\Delta x)\}$ is approximately equal to $f_X(x)\Delta x$ not exactly equal the way you have it (the approximation improving as $\Delta x$ approaches $0$), and you should include the proviso that the result holds only if $f_X(x)$ is continuous at $x$. Commented Jan 9, 2013 at 0:07
• It was useful . Commented Jan 9, 2013 at 8:10
• An explanation, why this notation is widely used and not odd, can be found here: math.stackexchange.com/a/599238/238307 It basically stems from the definition of the probability distribution as being the push-forward measure of $\mathbb P$ under the random variable $X$.
– wueb
Commented Jun 11, 2022 at 13:12

The notation $\mathbb P(X\in\mathrm dx)=f_X(x)\mathrm dx$ is odd, even as a shorthand, since $\mathrm dx$ can only mean an infinitesimal interval near $0$ and one wants to indicate an interval at $x$ or around $x$. The notation $\mathbb P(X=x)=f_X(x)$ is of course absurd since $\mathbb P(X=x)=0$ when $f_X$ exists. The notation $\mathbb P(X\in(x,x+\mathrm dx))=f_X(x)\mathrm dx$ is fine (and widely used). Personally I would not be too picky about using $(x,x+\mathrm dx)$ or $[x,x+\mathrm dx)$ as the infinitesimal interval here, since, anyhow, these are just semi-rigorous shorthands.

• For my benefit, could you clarify what exactly $\mathbb{P}(X\in(x,x+dx))$ means? Does it not mean anything on its own? Is it just used for shorthand in equalities, for example, is $\mathbb{P}(X\in(x,x+dx))=g(x)dx$ shorthand for $$\mathbb{P}(X\in(a,b))=\int_a^bg(x)dx$$ for any $a<b$?
– jkn
Commented Sep 5, 2013 at 15:25
• @jkn Your interpretation is fine.
– Did
Commented Sep 5, 2013 at 15:30
• – wueb
Commented Sep 14, 2022 at 18:17

The notation $F_X$ is very widely used to represent the cumulative distribution function (cdf) of a random variable $X$, and is defined as $F_X(x)=P[X\le x]$. This is valid whether or not $X$ is absolutely continuous. The probability density function, $f_X$, is only well-defined when $X$ is absolutely continuous, in which case $F_X$ is differentiable and $f_X=F'_X$. In this case only, your shorthand is legitimate, because $$f(x)=F'(x)=\lim_{\Delta x\rightarrow 0}\frac{F(x+\Delta x)-F(x)}{\Delta x}=\lim_{\Delta x\rightarrow 0}\frac{1}{\Delta x}P[X > x \wedge X\le x+\Delta x].$$ In terms of the cdf, the probability that $X$ takes on a particular value $x$ is $$P[X=x]=P[X\le x]-\lim_{y\rightarrow x-}P[X\le y]=F_X(x)-F_X(x-),$$ i.e., the difference between $F_X$'s value at $x$ and its limiting value approaching $x$ from below. Of course, this can only be nonzero if $F_X$ has a discrete component.

• That $X$ is absolutely continuous does not make $F_X$ differentiable hence the displayed series of identities involving $F'(x)$ needs qualification.
– Did
Commented Jan 9, 2013 at 6:35
• @did: You're right; absolutely continuous only means differentiable almost everywhere. So $f_X=F'_X$ where $F_X$ is differentiable, and can be assigned any value (say, $0$) at the remaining points, which have measure $0$. Commented Jan 9, 2013 at 7:43