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.


1 Answer 1


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)

Example 2

  • $\begingroup$ I'll try to use Chebyshev bounds but I would have to see how they can help with my original problem. I was hoping that getting the probability of a value under one set of moments I could get a quantile using a different set. @Glen_b $\endgroup$
    – jako
    Feb 4, 2014 at 8:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.