Let's say we have four moments $(\mu_1, \mu_2, \mu_3, \mu_4)$ of a probabilty distribution of a random variable $X$ and the goal is to get the probability $\rm{P}(X \leq t)$ for a certain value of $t$.
How is possible to use the moments to approximate such probabilty? Is there a way to use the moments in a Taylor expansion of the density?
Assume that $X$ has density which, at least, is four or five times differentiable, so the Taylor expansion makes sense.
Generally speaking, moments, even lots of them, may only fairly weakly pin down a density.
You can get Chebyshev-like inequalities from more moments than the first two, but even adding integer moments of all orders doesn't necessarily pin it down as much as one might hope (that link discusses bounds with all positive integer moments specified).
However, if the complete set of moments corresponds to a finite MGF in an open neighborhood of zero, then they will pin down the distribution. Without that, they don't.
More generally, two different distributions can have identical moment sequences (all moments equal) and still be quite different.
For two examples:
Example 1(a), $\, $ Example 1(b)
