Reading and interpreting a notation in probability mass function

Question

I'm studying a certain topic in probability until I came up in probability mass function in wikipedia. Then I saw the notation below but I don't understand it.

$$ f _ { X } ( x ) = \operatorname { Pr } ( X = x ) = P ( \{ s \in S : X ( s ) = x \} ) $$

How can I interpret it?

Answer 1

Actually it is a chain of two definitions:

• $\operatorname { Pr } ( X = x )$ is a symbol defined based on the given sample space $S$: $\operatorname { Pr } ( X = x ):= P ( \{ s \in S : X ( s ) = x \} )$
• The probability mass function $f _ { X } ( x )$ is now in turn defined by $f _ { X } ( x ) := \operatorname { Pr } ( X = x )$

That way you get a random variable defined only on a discrete set of real values where each value gets assigned a certain probability. This assignment is exactly the probability mass function.

Btw.: Instead of $\operatorname { Pr } ( X = x )$ you will see very often just $P(X=x)$.

p.s.:

You may play this through with the simple example of throwing two dice and you define $X$ as the sum of the points shown: Find $S$, then check out what $P(X=x)$ for $x=2,\ldots, 12$ is and then find $f_X(x)$.