IANAS, but I'll try an answer.

The problem is knwon as density estimation. A first option (which you probably will not find satisfactory) is an average of Kronecker deltas centered at the sampled points. Or you can replace the deltas with Gaussians or other shapes (kernel estimation).

As far as I know, in practice it is more common to assume a fixed (parameter-dependent) distribution and fit its parameters to the observed points with techniques such as maximum likelihood.

If you just need to find mean and variance, sample estimators are the traditional choice, and they do not go through the distribution. Their statistical properties are well studied.

IANAS, but I'll try an answer.

The problem is known as density estimation. A first option (which you probably will not find satisfactory) is an average of Kronecker deltas centered at the sampled points. Or you can replace the deltas with Gaussians or other shapes (kernel estimation).

As far as I know, in practice it is more common to assume a fixed (parameter-dependent) distribution and fit its parameters to the observed points with techniques such as maximum likelihood.

If you just need to find mean and variance, sample estimators are the traditional choice, and they do not go through the distribution. Their statistical properties are well studied.

The problem is knwon as density estimation. A first option (which you probably will not find satisfactory) is an average of Kronecker deltas centered at the sampled points. Or you can replace the deltas with Gaussians or other shapes (kernel estimation).

As far as I know, in practice it is more common to assume a fixed (parameter-dependent) distribution and fit its parameters to the observed points with techniques such as maximum likelihood.

IANAS, but I'll try an answer.

The problem is knwon as density estimation. A first option (which you probably will not find satisfactory) is an average of Kronecker deltas centered at the sampled points. Or you can replace the deltas with Gaussians or other shapes (kernel estimation).

As far as I know, in practice it is more common to assume a fixed (parameter-dependent) distribution and fit its parameters to the observed points with techniques such as maximum likelihood.

If you just need to find mean and variance, sample estimators are the traditional choice, and they do not go through the distribution. Their statistical properties are well studied.