Cumulative Distribution of double exponential

[Harashima Senju]

How do you find the CDF of a double exponential distribution, I'm having problems with removing the absolute x

 

[Hemant Rupani]

How do you find the CDF of a double exponential distribution, I'm having problems with removing the absolute x

Double exponential has same probability distribution for +ve & -ve sides( I.e. symmetry)
So first take CDF of -ve side of distribution.
Then, 0.5 + CDF of +ve side of distribution

 

[nad07]

In April 2011 the CDF for the double exponential distribution was given as

0.5exp(lambda_x) for x<0 and
0.5(1-exp(-lambda_x)) + 0.5 for x >=0

How was this obtained?

 

[Katherine Young]

You're given: \[g(x)= \frac{1}{2}\lambda e^{-\lambda\mid x\mid }\]

So for \[x\leqslant 0\]

\[g(x)=\frac{1}{2}\lambda e^{\lambda x}\]

We need to integrate to find the CDF:

So for \[ x\leqslant 0\]

\[G(x)=\int_{-\infty }^{x}\frac{1}{2}\lambda e^{\lambda t}dt = \frac{1}{2}e^ {\lambda x}\]

Now for \[x\geq 0\]

\[G(x)=P(x\leq 0)+\int_{0}^{x}\frac{1}{2}\lambda e^{-\lambda t}dt=\frac{1}{2}+\left [ -\frac{1}{2} e^{-\lambda t}\right ]_{0}^{x}=1-\frac{1}{2}e^{-\lambda x}\]

 

[Harashima Senju]

You're given: \[g(x)= \frac{1}{2}\lambda e^{-\lambda\mid x\mid }\]

So for \[x\leqslant 0\]

\[g(x)=\frac{1}{2}\lambda e^{\lambda x}\]

We need to integrate to find the CDF:

So for \[ x\leqslant 0\]

\[G(x)=\int_{-\infty }^{x}\frac{1}{2}\lambda e^{\lambda t}dt = \frac{1}{2}e^ {\lambda x}\]

Now for \[x\geq 0\]

\[G(x)=P(x\leq 0)+\int_{0}^{x}\frac{1}{2}\lambda e^{-\lambda t}dt=\frac{1}{2}+\left [ -\frac{1}{2} e^{-\lambda t}\right ]_{0}^{x}=1-\frac{1}{2}e^{-\lambda x}\]

I managed to solve it exactly the same way, thanks

 

[howdyrathore]

Katherine, can we use the alternate process mentioned in the CMP core reading given below?
Question: Generate a random variate X from the double exponential distribution with density function:

Solution: The density f is symmetric about 0; we can therefore generate a variate Y having the same distribution as |X| and
set X = + Y or X = - Y with equal probability. Now the density of |X| :

Algorithm:

The simulated numbers obtained are different in this process than the process you mentioned where we take F(x) values for x<0 and x>0. Are both methods allowed?

 

[John Lee]

Yes for generating values. Both methods would have scored full marks.

 

[Howard O'Connor]

What is the double exponential distribution? Where is it defined in the Core Reading?
Thank you

 

[John Lee]

It's an example in Chapter 14.