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I am trying out the first probability problem at this link which looks something like this:

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I would like to ask what is the difference between $\mathbf P(...)$ and $p_{...}(...)$? Aren't they both just probability? Why bother using different symbols? Thanks.

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  • $\begingroup$ I think $p()$ is the joint density, whereas $\mathbf P()$ refers only to $Y$. Is just to distinguish them as we use $Y$ for the r.v. and $y$ for the particular realization. $\endgroup$ – Patricio Feb 14 at 8:39
  • $\begingroup$ In this case $p_{X,Y}(x,y) = \mathbf P(X=x, Y=y)$ so you might want to say they are equivalent. Here $p_{X,Y}(x,y)$ takes the form of a function of $x$ and $y$ while $\mathbf P(X=x, Y=y)$ takes the form of the probability of an event, and you might want to draw that distinction. $\endgroup$ – Henry Feb 14 at 8:39
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The probability of an event A is written as $P(A)$, $p(A)$, or $Pr(A)$.

I'm recommending on this answers for your question:

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$\mathsf P(X=x)$ is the probability for the event of $X=x$.

$p_{\small X\!}(x)$ is the probability mass function of random variable $X$, measured at $x$.

Okay, yes, these are the same thing.

However, while we may generally measure the probability of various types of events, the probability mass function (pmf) of a random variable is something of special interest. So we have a special representation for it.

It describes the distribution of the random variable and can be used to derive various properties of the distribution. The mean, the variance, et cetera.

[Also $p_{\small X\!}(x)$ takes up slightly less typesetting on the page than $\mathsf P(X=x)$ , so can save space in long formulae.]

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