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Clarifying: "two probability distributions" -> "two distinct probability distributions" |
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If ''X'' is a [[vector space|vector]]-valued random variable, one takes the argument ''t'' to be a vector and ''tX'' to be a [[dot product]].
A characteristic function exists for any random variable. More than that, there is a [[bijection]] between cumulative probability distribution functions and characteristic functions. In other words, two distinct probability distributions never share the same characteristic function.
Given a characteristic function φ, it is possible to reconstruct the corresponding cumulative probability distribution function ''F'':
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