Laplace distribution

Quote:

Originally Posted by acasas4

The Laplace distribution, also known as a double exponential, has a pdf given by:
http://s3.amazonaws.com/answer-board...6347454136.gif
Find the theoretical mean and variance of a laplace distribution. (Hint: Integrals of absolute values should be done as a positive and negative part, in this case, with limits from -∞ to μ and from μ to ∞.)
I am not sure how to even go about this problem. I think you would have to set it as an integral using the following formula for the expected value of the mean: (lambda would be a constant)
http://s3.amazonaws.com/answer-board...3637714763.gif
http://s3.amazonaws.com/answer-board...6226437031.gif

If this is correct how would you solve this integral?

Thanks

You're expected to know, and the hint explicitly directs you to this, that

$\displaystyle f(x) = \frac{\lambda}{2} e^{-\lambda (x - \mu)}$ if $\displaystyle x - \mu > 0 \Rightarrow x > \mu$

$\displaystyle f(x) = \frac{\lambda}{2} e^{-\lambda (\mu - x)} = \frac{\lambda}{2} e^{\lambda (x - \mu)}$ if $\displaystyle x - \mu < 0 \Rightarrow x < \mu$

and break up the integral accordingly.