is f({Y|X}) equal to f(Y)|X?

An apparently meaningless question on X validated last week, where the confusion between conditional density and the transform of a conditional variate led viewers astray… The question was in fact whether transform and conditioning are exchangeable, i.e., whether or not the transform by f of the random-variable-Y-conditional-on-X is distributed as the tranform-by-f-of-the-random-variable-Y conditional on X. Which appears as a conditional version of the (irritatingly called) law of the unconscious statistician…

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