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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ 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|0DKRR22AEQ]].\nThe law of $X$, also called the distribution of $X$, is the function $P_X \\colon \\mathcal{E} \\to [0, 1]$ with\n$$P_X(B) = P(X^{-1}(B)) = P(X \\in B) \\quad \\text{for } B \\in \\mathcal{E}.$$\n\n## Remarks\n\n$P_X$ is a [[probability distribution|0QB913WM5E]] on $(E, \\mathcal{E})$. It is the image measure, or pushforward, of $P$ under $X$, and one also writes $P \\circ X^{-1}$ or $X_* P$.\nNotation: $X \\sim \\mu$ means that the law of $X$ is $\\mu$.\nTwo 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 $1/2$.\n\n## Explanations\n\n### L1 human\n\nThe law of $X$ forgets the sample space and keeps only the chances of the values of $X$.\nFor the sum of two dice, the law says that $7$ has probability $6/36$, $2$ has probability $1/36$, and so on.\n",
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