First and second moments of truncated laplace distribution I'm trying to estimate a distribution that looks like a truncated Laplace distribution. However, I can't find closed-form expressions of its first and second moments. I'm expecting closed-form ones as in truncated normal distribution (see https://en.wikipedia.org/wiki/Truncated_normal_distribution). 
Have you guys found it before? 
 A: Given:  $X \sim \text{Laplace}(\mu, \sigma)$ where $-1 < \mu <1$, with pdf $f(x)$:

(source: tri.org.au) 
Then, the (doubly truncated) conditional density, truncated above at 1, and below at -1, is:
$$g(x) \; = \; f(x \;\big|-1<X< 1) \; = \; \frac{f(x)}{P(-1<X<1)} $$
Here, $P(-1<X<1)$ is given by:

(source: tri.org.au) 
Then, the doubly truncated pdf $g(x)$ is:

(source: tri.org.au) 
Here is a plot of the doubly truncated Laplace pdf $g(x)$, given some different paramater combinations:

(source: tri.org.au) 
The OP seeks the first and second moments of $X$, when X has doubly truncated pdf $g(x)$. The first moment is $E_g[X]$:

(source: tri.org.au) 
and the second moment is $E_g[X^2]$:

(source: tri.org.au) 
where I am using the Expect function from the mathStatica package for Mathematica to automate the calculations.
A: There is a closed form for the cdf, that is just a linear scale on the non-truncated cdf. In particular if $X \sim F_X$,  and $Y$ follows the same distribution but truncated on the interval $[a;b]$, then the cdf for $Y$, $F_Y(\cdot)$ is given by $F_Y(y) = \frac{1}{F_X(b)-F_X(a)} F_X(y)$. 
To get, say, the first moment, 
$$\mathbb{E}(Y) = \int_{-\infty}^{\infty} yf_Y(y) dy = \frac{1}{F_X(b)-F_X(a)} \int_a^b yf_X(y) dy.$$ 
The problem is in general, that we don't know what happens to the last integral when we restrict it to be on $[a;b]$. 
On the truncated normal distribution
I was imprecise earlier: Here is the derivation for the truncated normal: 

As you can see, the crucial step is that the derivative of the normal density is equal to minus the inside times the normal density. In this sense, there is a type of linear invariance of the normal density and without it, this derivation would not work out to be a simple form. 
