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  "body": "\n## Definition\n\nLet $X$ and $Y$ be random variables on one probability space.\nDiscrete case: let $X$ and $Y$ be discrete and let $x$ satisfy $P(X = x) \u003e 0$. The conditional distribution of $Y$ given $X = x$ has the [[probability mass function|0WF4EJP91W]]\n$$p_{Y \\mid X}(y \\mid x) = P(Y = y \\mid X = x) = \\frac{P(X = x, \\, Y = y)}{P(X = x)}.$$\nContinuous case: let $(X, Y)$ have a joint density $f_{X, Y}$, and let $f_X$ be the [[marginal|0AXP2Z5N9H]] density of $X$. For each $x$ with $f_X(x) \u003e 0$, the conditional distribution of $Y$ given $X = x$ has the [[density|04YG690N5V]]\n$$f_{Y \\mid X}(y \\mid x) = \\frac{f_{X, Y}(x, y)}{f_X(x)}.$$\n\n## Remarks\n\nIn the discrete case, $p_{Y \\mid X}(\\cdot \\mid x)$ is the law of $Y$ under the [[conditional probability|0FJB06AD2P]] $P(\\cdot \\mid X = x)$.\nIn both cases, the joint law is the marginal times the conditional: $f_{X, Y}(x, y) = f_X(x) \\, f_{Y \\mid X}(y \\mid x)$.\nIf $X$ and $Y$ are independent, the conditional distribution of $Y$ given $X = x$ is the law of $Y$, for each such $x$.\nIn general measure theory, one defines it as a regular conditional distribution. This is a probability kernel $\\kappa$ with $P(X \\in A, Y \\in B) = \\int_A \\kappa(x, B) \\, \\mu_X(dx)$.\n\n## Explanations\n\n### L1 human\n\nThe conditional distribution answers the question: \"If I know $X$, how is $Y$ distributed?\"\nFor a person of height $180$ cm, it describes the possible weights of that person and their chances.\n",
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