Moments of Laplace distribution I am a newbie in stat. I am working on the Laplace distribution for my algorithm. 


*

*Could tell me the first what the four moments of the Laplace distribution are? 

*Does it have infinite tail like the Cauchy distribution? 

*What is the empirical rule?

 A: Here is a quick check using a symbolic algebra package ...
Let $X \sim \text{Laplace}(\mu, \sigma)$ with pdf $f(x)$:

Then, the first 4 raw moments $E[X^i]$ are given by:

where I am using the Expect function from the mathStatica package for Mathematica. 
It is worth noting that the $3^\text{rd}$ and $4^\text{th}$ raw moments are different to those given in the answer above.
A: It's been awhile and looks like this hasn't been answered. I'll provide one and hopefully we can mark this as correct. I'll answer in order the questions asked using the parameterization of the wikipedia page
$$f(x\mid\mu,b)= \frac{1}{2b} \exp \left( -\frac{|x-\mu|}{b} \right), x\in \mathbb{R}. $$

*

*For the case $\mu=0$, the first four moments are: $$\mathbb{E}(X)=0, \mathbb{E}(X^2)=2b^2 + \mu^2, \mathbb{E}(X^3)=0, and\ \mathbb{E}(X^4) = 24b^4.$$ As whuber indicates in a comment you can related a non-central random variable $Y$ via a binomial expansion of $Y^k=(Xb+\mu)^k$. The value $\mu=0$ is often chosen to simplify the calculation and to build up to the solution.


*The Laplace have infinite tails like the Cauchy, the support is $x \in (-\infty, \infty)$.


*For the empirical rule, I'm assuming the OP is using the shorthand for the probability of observations within $\sigma$ of the mean, $\mu$,  $2\sigma$ of $\mu$ and $2\sigma$ of $\mu$ respectively. These probabilities are: (0.75688, 0.94089, 0.98563) to 5 significant digits, respectively.
A couple of different ways to calculate the expected values are:
a. Direct integration (usually split the integral at the point $\mu$ where the sign changes.
b. Differentiate the moment generating function $m$ times and set $t=0$ to get the $m$th moment.
c. Formulate the Laplace random variable (r.v.) as a scale mixture of Normal and Exponential random variable. Then use conditional expectations.
Note approach (c) is only easier than (a) if you know the moments of Normal and Exponential random variables-or can calculate them easier than directly calculating the moments of the Laplace distribution
