# Finding $\mathbb{E}|X|$ for variance-gamma random variable

Assume $$X \sim f_{X}(n,\mu, \sigma)$$ is a Variance-Gamma random variable. The density function involves a modified Bessel function, therefore is not that trivial to handle.

I'm looking for $$\mathbb{E}|X|$$ (and, by extension, $$Var(|X|)$$)

Looking at this straightforwardly, it should hold that $$\mathbb{E}|X| = \int_{-\infty}^{\infty} x dF_{|X|}$$ where $$F_{|X|}(t)=\mathbb{P}(|X|\leq t) = \mathbb{P}(-t \leq X \leq t)$$.

Without deriving the direct expression for $$F_{|X|}(t)$$, can we arrive at $$\mathbb{E}|X|$$ through integration? Or would this be easier through characteristic functions?

• The easiest way is probably to go through the mixture representation (i.e. condition, use known results for the normal distribution, then use the law of total expectation/variance). – Chris Haug Apr 20 at 0:21

This is just an extended comment.

You might want to try a symbolic computation program such as Mathematica, Maple, or MATLAB. Here are the results for the mean and variance using Mathematica. (And maybe there is some simplification available?)

dist = TransformedDistribution[Abs[x],
x \[Distributed] VarianceGammaDistribution[\[Lambda], \[Alpha], \[Beta], 0]];
mean = Mean[dist] var = Variance[dist] • Wow, that's a bit more complicated expression than I initially expected. Thanks for the code, will play around with it. – runr Apr 20 at 12:25