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I have a lecture note saying "In our class the symbol $p_X (x)$ reffers to the whole of the probability density function e.g. the same as $p_X=p_X (\cdot)=\{p_X(x):x\in \mathbb X \}$". I tried following this post to understand the notation $p_X(\cdot)$. When is it ok to use it and how do I know wht is it reffering to?

havakok
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    When you want to refer to a function rather than to the value it takes for a given argument. Nothing specific to probability. For example, $\sin(\cdot)$ and $\sin$ are synonymous. – Did Oct 28 '17 at 09:18
  • Got it, thanks. – havakok Oct 28 '17 at 09:21
  • Personally I like to write $x \sim p_X(\cdot)$ to emphasize that I will only use $p_X$ to sample from while I use $p(x)$ as shorthand for the pdf (in closed form) with x as input variable. Notation in this field is a complete mess though. – Anne van Rossum Oct 11 '20 at 11:38

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Let $f$ be the absolute value function on $\mathbb{R}$. In other words,

$f:\mathbb{R}\rightarrow\mathbb{R},\ f(x) = |x|$. Everyone is familiar with the absolute value, so it's annoying having to define an entirely new function - so why don't we use the same notation as the absolute value?

Instead of $f:\mathbb{R}\rightarrow\mathbb{R}$, we can say $|\cdot|:\mathbb{R}\rightarrow\mathbb{R}$. In this case, you see that $x$ gets mapped to $|x|$. The "dot" is just a placeholder, saying "this is where you put the $x$".

In the same way, if you have some probability function $p_X$, you might want to emphasise that it's a function rather than a constant. If you write $p_X(\cdot)$, it's very clear that it's a function that takes inputs where the "dot" is, i.e. $x$ gets mapped to $p_X(x)$.

Note that it's generally not good practice to write $p_X(x)$ for an entire function, because it should really be a single point in the image of the function. (But in your class they're saying $p_X(x) = p_X(\cdot) = p_X$ I guess.)

Harambe
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  • "But in your class they're saying $p_X(x) = p_X(\cdot) = p_X$ I guess" Well, it seems the situation "in their class" is exactly the opposite in the sense that "they" take care to make a distinction between the function $p_X$ and one of the values $p_X(x)$. For once, let us rejoice that "they" do... – Did Oct 28 '17 at 09:28
  • @Did I have left out, the distinction is petween the case of a discrete variable $x$, where $p_X$ is the value of the probability vs. a conyinious variable $x$, where $p_X$ is the PDF. – havakok Oct 28 '17 at 09:37
  • @havakok The distinction discrete/continuous seems irrelevant here. – Did Oct 28 '17 at 11:30