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Assume $X$ is a discrete-valued random variable. Often in literature the same notation is used for the probability $Pr(X = x)$ as for the probability mass function $P(X)$. Clearly, this is because both entities represent probabilities: $Pr(X = x)$ represents probability directly, whereas $P(X)$ is a probability-valued function.

Under what circumstances does the distinction between the two objects $Pr(X = x)$ and $P(X)$ become important, or even crucial? Or, alternatively, are there theoretical reasons why it is always ok to use the same notation to refer to both?

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They are referring to two different things as you already said. So the distinction is important. The function is different than the output of the function.

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  • $\begingroup$ Do you have a specific example in mind, which demonstrates the need to make this distinction? $\endgroup$ – funklute Feb 27 '19 at 16:01
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Probability is a number. Probability mass function is a function that assigns probabilities to some numbers. So the distinction is simple: if it is a function, then it is a probability mass function. This will always follow from the context.

The simplest example of probability mass function is Bernoulli distribution, that assigns probabilities to values of $X$,

$$\begin{align} f(x) &= \begin{cases} p & \text{if }x=1, \\[6pt] 1-p & \text {if }x=0.\end{cases} \\ &= p^x \,(1-p)^{1-x} \end{align}$$

It assigns probabilities $\Pr(X=1)=p$ and $\Pr(X=0) = 1-p$ to $X \in \{0,1\}.$

As about notation, you can write probabilities as $P(X=x)$, $p(X)$, $\Pr(X=x)$, $\mathrm{Prob}(X=x)$, $\mathbb{P}(X)$ and many more different notations, in italics, or not, sometimes in blackboard font, sometimes other font, etc. There is no strict rules about it, the notation is used differently by different authors and editors. Notation in mathematics is in many cases ambiguous and flexible, and uses many shorthand's, so the meaning needs to be understood given the context.

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  • $\begingroup$ So I'm actually quite happy with the distinction between a function and a probability, and I'm also happy with how this notation is loosely used in literature. What I'm more interested in, is whether there are "edge cases" where confusing the two kinds of objects can lead to faulty arguments and/or conclusions, e.g. when going deeper into probability and measure theory? $\endgroup$ – funklute Feb 28 '19 at 15:04
  • $\begingroup$ @funklute I can't think of any "edge cases". Probability is of something specific. PMF is a function that describes a random variable. Example of Bernoulli PMF is given, where as soon as you start talking about function describing probabilities of the the two possible events, it is PMF. $\endgroup$ – Tim Feb 28 '19 at 15:25

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