Demystification of Probability Notation


It is quite common to abuse notation when working with probability distributions. E.g., when two random variables $X$ and $Y$ are independent it is common to see $$ \begin{align} & p(x, y) = p(x)p(y). \end{align} $$

When presented with this notation it can be difficult to understand where the random variable ($X$) is, the realization of this random variable ($x$), and that $p(x)$ and $p(y)$ are two different functions. A proper way to write the equation above is $$ \begin{align} & p_{X,Y}(x, y) = p_X(x)p_X(y), \end{align} $$

where it becomes clear that $p_{X,Y}$, $p_X$ and $p_Y$ are three different functions.

The notation gets more confusing when computing expectations. The expectation operator is applied to a function of a random variable. In many works one reads $$ \begin{align} &\mathbb{E}_{p(x)}\left[ f(x) \right] = \int p(x)f(x)\mathrm{d}x. \end{align} $$

Again we assume that $p(x) \equiv p_X(x)$ and that $f(x)$ on the LHS is not a function of the realization but of the random variable. An ambiguous way to write the above would be $$ \begin{align} &\mathbb{E}\left[ f(X) \right] = \int p_{X}(x)f(x)\mathrm{d}x. \end{align} $$


It is OK to abuse the notation when we know exactly where random variables and their realizations are. But in case we get confused, it can be easier to write everything explicitly for once.

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