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  "body": "\n## Definition\n\nLet $(E, \\mathcal{E})$ be a set with a σ-algebra.\nLet $X$ be a random variable on a probability space $(\\Omega, \\mathcal{F}, P)$ with values in $E$.\nLet $Y$ be a random variable on a probability space $(\\Omega', \\mathcal{F}', P')$ with values in $E$.\n$X$ and $Y$ are identically distributed if they have the same [[law|0TST4MHA3A]]: $P(X \\in B) = P'(Y \\in B)$ for each $B \\in \\mathcal{E}$.\nOne writes $X \\overset{d}{=} Y$.\n\n## Remarks\n\nFor real random variables, this holds if and only if $X$ and $Y$ have the same cumulative distribution function.\nIdentically distributed random variables need not be equal. For a fair coin, the indicator $X$ of heads and the indicator $1 - X$ of tails are identically distributed, but they are never equal.\nIf $X \\overset{d}{=} Y$ and $g$ is a measurable function, then $g(X) \\overset{d}{=} g(Y)$.\n\n## Explanations\n\n### L1 human\n\nTwo random variables are identically distributed when they behave the same way statistically.\nThe results of two different fair dice are identically distributed, although the dice show different numbers in most throws.\n",
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