Laplace distribution and, generally, interpreting an undefined moment I'm studying the distributional properties of a laplace distribution, and I'm trying to get some intuition beyond plotting the distribution of what it means to have an undefined moment.
In wikipedia you can see that the mgf is only defined for $|t| < 1/b$ so as the variance of the laplace distribution increases to 1, you lose all moments including the mean. Does this matter? What is the intuition? For example the fourth moment can blow up, but the distribution will still look generally okay. What is the benefit of having a defined moment if you have the distribution?
If I have some data that fits the laplace distribution fits well with a very high b, should I be concerned? If for two data sets where b is close to 1 in one data set, but smaller in another, am I more confident in the fit to the data set with a b that generates more moments?
Any thoughts would be greatly appreciated. And if I'm thinking about this the wrong way let me know.
 A: I was incorrectly using the moment generating function which led to my misunderstanding of the Laplace distribution. 
The moment generating function is  $M_X(\theta) = \text{E}(e^{\theta X})$. 
When you use that to find the $n^{\text{th}}$ moment, you take the $n^{\text{th}}$ derivative at $\theta=0$: 
$$\frac{d^{n}(M_X(\theta))}{d(\theta)^{n}} |_{\theta=0}\quad\text{.}$$ 
If you see my note above there is a proof using Taylor series expansion of $E(e^{\theta X})$. When you take the n-th derivative the leading term will not have a $\theta$ in it, but higher order terms will have $\theta$. This allows you to set $\theta=0$ and use the moment generating function to produce moments.
So for the Laplace we have  $E(e^{\theta X}) = e^{\mu\theta}/(1-b^{2}\theta^{2})$ (from Wikipedia)
$E(X) = d^{1}(M_X(\theta))/d(\theta)^{1} = (e^{\theta\mu} (\mu + b^2 \theta (2 - \theta \mu)))/(-1 + b^2 \theta^2)^2$
if you evaluate this for $\theta=0$, then you get $E(X) = \mu$ as expected.
Now the second part of my question is trying to understand undefined moments. The implication of an undefined moment means that trying to estimate the parameters of the distribution by matching moments will generally require more advanced techniques (such as maximizing log-likelihood).
There is a good discussion about this for the Cauchy distribution which does not have defined moments see http://en.wikipedia.org/wiki/Cauchy_distribution
As an added thought, in Python there is a symbolic algebra package called sympy that makes evaluating moments very simple using symbolic algebra. There are simple formulas to convert non-central moments to central moments, allowing you to calculate skewness and kurtosis fairly easily for many distributions.
