Random Variables & the Probability Mass Function

A random variable (Definition 12.1) is neither a varying number nor random on its own — it is a function $X:\Omega\to\mathbb{R}$ that attaches a real number $X(\omega)$ to every outcome $\omega$ of an experiment. The randomness lives entirely in which outcome $\omega\in\Omega$ the experiment produces; once $\omega$ is fixed, the value $X(\omega)$ is completely determined. By convention random variables wear uppercase letters $X,Y,Z$, while $X(\omega)$ denotes the value at a sample point. For a pair of dice, $\Omega=\{(i,j):i,j\in\{1,\dots,6\}\}$, and we may define $X_1(i,j)=i$ (first die), $X_2(i,j)=j$ (second die), and $S(i,j)=X_1(i,j)+X_2(i,j)=i+j$ (the sum); for the outcome $(5,1)$ these give $X_1=5$, $X_2=1$, $S=6$. A random variable thus repackages raw outcomes into the number we actually care about — a total, a payoff, a count — converting an experiment about dice faces into one about sums.

Because $X$ is a function, we can ask how likely it is to land in a target set. For a subset $B\subseteq\mathbb{R}$, the shorthand $\{X\in B\}$ (read '$X$ lies in $B$') names the event consisting of every outcome whose $X$-value falls in $B$: $\{X\in B\}=\{\omega\in\Omega:X(\omega)\in B\}.$ Since this is a genuine subset of $\Omega$, it is an event in $\mathcal{F}$ and carries a probability $\mathbb{P}\{X\in B\}$ inherited from the underlying probability space. Useful special cases are $\{X=k\}$ (take $B=\{k\}$) and $\{X\le k\}$ (take $B=(-\infty,k]$). The probability distribution of $X$ (Definition 12.2) is the entire collection of these numbers $\mathbb{P}\{X\in B\}$ as $B$ ranges over subsets of $\mathbb{R}$ — a complete description of how $X$ spreads its probability along the real line. For the dice sum, $\{S=8\}$ collects $(2,6),(3,5),(4,4),(5,3),(6,2)$, so $\mathbb{P}\{S=8\}=\tfrac{5}{36}$.

A random variable is discrete (Definition 12.3) when its range — the set of values it can take — is finite or countably infinite, say $\{k_1,k_2,\dots\}\subseteq\mathbb{R}$, and these values carry all the probability: $\sum_i \mathbb{P}\{X=k_i\}=1.$ The values $k$ with $\mathbb{P}\{X=k\}>0$ are the possible values of $X$. Discreteness is exactly what makes exact, by-hand computation possible: a discrete $X$ places all of its probability on a list of isolated points, so any probability about $X$ is a sum over those points, never an integral. The sum of two dice ($S\in\{2,\dots,12\}$) and the wealth change ($W\in\{-1,1,3\}$) are discrete with finite range; a quantity that can in principle grow without bound (such as the number of tosses until the first head) is discrete with countably infinite range. Continuous random variables — whose probability is smeared over intervals and found by integration — are the subject of a later lesson.

For a discrete $X$ the whole distribution is captured by one function. The probability mass function (p.m.f., Definition 12.4) is $p(k)=\mathbb{P}\{X=k\},$ defined on the possible values of $X$: it records how much probability 'mass' sits at each point. Knowing the p.m.f. is knowing everything, because for any set $B\subseteq\mathbb{R}$ $\mathbb{P}\{X\in B\}=\sum_{k\in B}p(k),$ so every probability about $X$ is read off by summing masses (for instance $\mathbb{P}\{X\le k\}=\sum_{j\le k}p(j)$). A function $p$ is a valid p.m.f. precisely when it meets two conditions mirroring the probability axioms: it is nonnegative, $p(k)\ge 0$ for every $k$ (a probability is never negative), and it has total mass one, $\sum_k p(k)=1$ (all the probability is accounted for). Any $p$ meeting both defines a legitimate discrete random variable; fail either and it is not a p.m.f. For example the first die has $p_{X_1}(k)=\tfrac16$ for $k=1,\dots,6$ (six masses of $\tfrac16$ summing to $1$), and the wealth game has $p_W(-1)=\tfrac12,\ p_W(1)=\tfrac16,\ p_W(3)=\tfrac13$.