1. ## laplace cdf integration and kurtosis

Hello,
given the laplace distribution. f(x)=0.5*e^(-|x|)
How do you integrate in order to find the cdf for x>0 (I have no problem finding the solution for x<0)

The solution is 1-0.5*e^(-x)

I would be very grateful for a detailed solution or a link to a website were I could find it.

2. ## Re: laplace cdf integration

Hello,

Well the best way is to see that if X follows a Laplace distribution, so does -X, since the pdf is symmetric over 0... Then for x>0, $P(X\leq x)=P(X-x)=P(X>-x)=1-P(X\leq -x)=1-F_X(-x)$

3. ## Re: laplace cdf integration

another thing: (I don't know whether I should start a new thread for this, I changed the title of the original post to make it clear that there is a 2nd part to the original question)

how do I derive the kurtosis of the laplace function using an integral of the pdf? I will have to integrate by parts, but how?

4. ## Re: laplace cdf integration

Originally Posted by Weezphili

another thing: (I don't know whether I should start a new thread for this, I changed the title of the original post to make it clear that there is a 2nd part to the original question)

how do I derive the kurtosis of the laplace function using an integral of the pdf? I will have to integrate by parts, but how?
Well the kurtosis is known to be $E\left[\left(\frac{X-\mu}{\sigma}\right)^4\right]$
Here, the "location" is $\mu=0$ and the "scale" is $b=1$. But we know that the variance is $\sigma^2=2b^2=2$, hence $\sigma=\sqrt{2}$

So the kurtosis will be $\frac12 E\left[X^4\right]=\frac12 \int_{\mathbb R} x^4 f_X(x) ~ dx=\frac14 \int_{\mathbb R} x^4 e^{-|x|} ~dx=\frac 12 \int_0^\infty x^4 e^{-x} ~dx$ (symmetry over 0).

You can do 3 (I think) successive integrations by parts or you can simply recognize the gamma function, which is much easier :P

5. ## Re: laplace cdf integration

Originally Posted by Moo
You can do 3 (I think) successive integrations by parts or you can simply recognize the gamma function, which is much easier :P
It's exactly with these integrations by parts that I am having trouble

6. ## Re: laplace cdf integration

Let u'=e^{-x} and v=x^4
Every time, let u'=e^{-x}