If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. Does anyone know an example of two different probability distributions with identical moments? The less pathological the better.
**Edit:** Is it unconditionally true if I specialize to discrete distributions?

And a related question: Suppose I ask the same question about Renyi entropies. Recall that the Renyi entropy is defined for all `a`

≥ 0 by

H_{a}(p) = log(∑_{j} p_{j}^{a})/(1-a)

You can define `a`

=0,1,∞ by taking suitable limits of this formula. Are two distributions with identical Renyi entropies (for all values of the parameter `a`

) actually equal? How "rigid" is this result? If I allow two Renyi entropies of distributions `p`

and `q`

to differ by at most some small ε independent of `a`

, then can I put an upper bound on, say, || p - q ||_{1} in terms of ε? What can be said in the case of discrete distributions?