Generating random nos based on 'k' moments How do I generate random nos based on say k moments? (no other constraints on support)
When k = 2, we generate random nos. from a normal distribution defined by the 2 moments. Can we generalize this idea to an arbitrary k.
A related question: Is normal distribution the only (known) distribution which is defined by just 1st 2 moments?
I wouldn't mind suggestions for books/links, instead of direct answers to these questions.
(Editing the question based on a couple of answers I got)
I'll try to provide the context in which I thought of this question. I have a uni variate sample and no more information about anything. Now, I want to try and simulate nos. from the distribution from which this sample was drawn. I could just find its 1st 2 moments, assume this sample to be normally distributed, and be done. But if I want do away with 'Normal' assumption, I can try to find it's first k moments and somehow draw from the distribution defined by these k moments.
I am interested in its theoretical underpinnings as well as practical implementation.
 A: There are many distributions that involve just one or two parameters and hence can be determined by just first or first and second moments.  For example the exponential distribution is defined by its rate parameter which can be determined from the first moment.
Other distributions that may involve three or more parameters may need higher order moments to be determined.  A method of estimation technique called the method of moments predates maximum likelihood and is based on solving equations that relate the model paremters to their moments.  The moments are estimated from the data and then the equations are used to get the parameter estimated.
If you want to generate random numbers from a specific distribution that involves several parameters, it may be possible to specify the distribution based on the first few moments.
A: Moments and parameters are two different things.  By using the limit k I presume you mean that the moments beyond moment k are all zero?  The exponential distribution is skewed, so its third moment is not zero.  If I recall correctly from studying moment generating functions, the Normal distribution is the only distribution in which moments beyond the second moment are all zero.
A: So you want to simulate from a distribution given only empirically by some sample!
Is there any reason to not use the bootstrap? Or you could make a kernel density estimation, and then simulate from that! (If you are using a normal kernel function, that would be like sampling (with replacement) from the empirical distribution (that is, bootstrapping) and then adding a normally distributed "fuss" to each sampled value.
Another idea, which might be useful if it is important that the si,ulation not be constrained to the support of the sample, is to calculate an empirical saddlepoint approximation and simulate from that. Some issues which arise is discussed in How does saddlepoint approximation work?
