Law of a random variable

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $(E, \mathcal{E})$ be a set with a σ-algebra, and let $X \colon \Omega \to E$ be a random variable. The law of $X$, also called the distribution of $X$, is the function $P_X \colon \mathcal{E} \to [0, 1]$ with $$P_X(B) = P(X^{-1}(B)) = P(X \in B) \quad \text{for } B \in \mathcal{E}.$$

Remarks

$P_X$ is a probability distribution on $(E, \mathcal{E})$. It is the image measure, or pushforward, of $P$ under $X$, and one also writes $P \circ X^{-1}$ or $X_* P$. Notation: $X \sim \mu$ means that the law of $X$ is $\mu$. Two random variables can have the same law but be different functions. For a fair coin toss, the indicator of heads and the indicator of tails both have the Bernoulli law with parameter $1/2$.

Explanations

L1 human

The law of $X$ forgets the sample space and keeps only the chances of the values of $X$. For the sum of two dice, the law says that $7$ has probability $6/36$, $2$ has probability $1/36$, and so on.