I was given in a set of lecture notes that

If a function $p_X(x)$ satsifies:

(a) $p_X (x) \geq 0$, $\forall x \in \mathbb{R}$

(b) $\sum _{x } p_X(x) = 1$

Then $p_X(x)$ is a probablity mass function.

The definitions I was given are as follows:

Discrete Random Variable:

A discrete random variable $X$ on a probability space $( \Omega, \mathcal{F}, \mathbb{P})$ is a function $X : \Omega \rightarrow \mathbb{R}$ such that

(a) $\{ w \in \Omega : X(w) = x \} \in \mathcal{F}$ for each $x \in \mathbb{R}$

(b) $Im(X) := \{ X(w): w \in \Omega \}$ is a finte or countable subset of $\mathbb{R} $

Probability Mass Function:

The probability mass function (pmf) of $X$ is the function $p_X : \mathbb{R} \rightarrow [0.1]$ defined by $$ p_X (x) := \mathbb{P}(X=x) $$

Maybe it seems trivial to you all, but may someone explain how does the statement follow from the definitions? Or maybe the statement is not true? Thank you so much!


(a) $\{ \omega \in \Omega : X(\omega) = x \} \in \mathcal{F}$ for each $x \in \mathbb{R}$

This is just saying that the subset of outcomes with X-measure equal to any real value, $x$, must be an element of the sigma-algerbra, $\mathcal F$ of the probability space.   It follows that we can P-measure each of these subsets.   That's what we mean by $\mathsf P(X{=}x)$ when the probability space is $(\Omega, \mathcal F, \mathsf P)$ and $X:\Omega\to\Bbb R$.

tl;dr $\mathsf P(X{=}x) \mathop{:=} \mathsf P(\{\omega{\in}\Omega:X(\omega){=}x\})\\ \therefore \forall x\in\Bbb R: \mathsf P(X{=}x) \geq 0$

Further, the non-empty subsets must partition the space.

(b) $\operatorname{Im}(X) := \{ X(\omega): \omega \in \Omega \}$ is a finite or countable subset of $\mathbb{R} $

This means that the values that are X-measures of all outcomes of the space constitute a set of discrete points.   Then the series of P-measures of these points is $1$ (since their corrosponding subsets partition the space).

tl;dr $\therefore \sum_{x\in\operatorname{Im}(X)} \mathsf P(X{=}x) = 1$


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