1. ## Laplace distribution

The Laplace distribution, also known as a double exponential, has a pdf given by:

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)

If this is correct how would you solve this integral?

Thanks

2. ## Re: Laplace distribution

Originally Posted by acasas4
The Laplace distribution, also known as a double exponential, has a pdf given by:

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)

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

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

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

and break up the integral accordingly.

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# mean of laplace dis. with proof.

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