Normalizing a continuous distribution I work at a help/tutoring center at my university.  Today a kid came in with this problem.  I've only studied math and haven't drifted into physics, but he had this problem:
Let $P(x)=Ne^{-\frac{|x|}{a}}$.  Then:  
(a)  Find $N$ such that $P(x)$ is properly normalized. 
(b)  Find $\langle x \rangle$ and $\langle x^2 \rangle $
Attempt at solution:
$1=\displaystyle\int_{-\infty}^{\infty}Ne^{-\frac{|x|}{a}}=2\int_{0}^{\infty}Ne^{-\frac{x}{a}}=2\int_{0}^{\infty}Ne^{-u}a\,du=2Na\int_{0}^\infty e^{-u}du=2Na.$  
So $N=\frac{1}{2a}$
I'm not sure how to do part (b).  The text, Robinett's Quantum Mechanics, was a bit beyond me.  Could I get some guidance on how to solve this problem...it seems really interesting.

For part a) you should check that the symmetry of $P(x)$ with respect to $x$ gives 
$$\int_{-\infty}^{\infty}Ne^{-|x|/a} dx=2\int_{0}^{\infty}Ne^{-x/a}dx$$
and then do a change of variable $\displaystyle u=\frac{x}{a}$
For part b) note that by definition $\displaystyle\langle f(x) \rangle=\int_{-\infty}^{\infty}f(x)Ne^{-|x|/a}dx$ and that both $x$, $x^2$ and $P(x)$ have defined symmetry with respect to $x$ and you can exploit that symmetry to know the value of some of them (in particular $\langle x \rangle$ without actually computing the integrals
