Is this true? Assume that, for all real numbers k, there exists I such that p)X(x|I) = f(x-k).

[FrederikDan]

Can the following be proven:

[imath]\textrm{Assume } \forall\,k\in\mathbb{R}\,\exists I:\;\,p_X(x\,|\,I)=f(x-k) [/imath]

then prove that it follows that for some unknown [imath]I[/imath] it is true that [imath]p_{K}(k) \equiv p_X(x)[/imath], meaning that [imath]p_K[/imath] and [imath]p_X[/imath] are the same distribution, where [imath]p_X(x)[/imath] denotes the probability distribution of the stochastic variable [imath]X[/imath] and [imath]I[/imath] denotes some information.

 

[blamocur]

Where does this problem come from, and is this an exact statement? You say "...for some unknown [imath]I[/imath]", but that [imath]I[/imath] is not a part of the remaining sentence. And then you introduce a new (stochastic?) variable [imath]K[/imath].