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Kyle Sutton 2022-07-10 Answered
Let ( Ω , F , P ) be a probability space and X : ( Ω , F ) ( R , F E ) a random variable that admits a density function f X : R R .
I have some questions about the information given by the cumulative distribution function of X ( F X : R [ 0 , 1 ]).
i) Does the CDF determine f X uniquely ?
ii) Does the CDF determine the law of X uniquely ?
iii) Can we find a subset of events that uniquely determine the CDF ?
For i), I would say that as F X ( t ) = t f X ( x ) d x, the density function is the derivative of the CDF and therefore the CDF can only define one density function. But I feel like maybe we could find a counterexample by considering Lebesgue measure and taking sets of the form (a,b) and [a,b].
For ii), as x R , P ( X = x ) = F X ( x ) F X ( x ) and P ( X x ) = F X ( x ), I am tempted to say that the law of X is indeed by definition determined uniquely by the CDF.
For iii), maybe we could take all events of the form { X ( a , b ) : a , b R } but I'm not quite sure about that, and particularly regarding the uniqueness.
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Answers (1)

razdiralem
Answered 2022-07-11 Author has 15 answers
By subtraction, the CDF F ( x ) := P ( X x ) determines the probabilities P ( X [ a , b ] ) for every a , b R . By an approximation theorem, such as Caratheodory's theorem, this information determines P ( X E ) for every E B ( R ). So (ii) is true.
(i) follows (though, as always, the density is unique only up to almost everywhere equality).

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