My notation:

X: Your double exponential variable

Y: a "regular" exponential variable with the same parameter (pdf = )

Answer

the pdf of X is symmetric about 0 so E(X) = 0

because of symmetry E(X^2) will be the same as for a regular exponential variable (check you can prove / understand this, your teacher will prboably expect you to show the integral rather than jsut saying "symmetry").

so we have (RESULT 1)

The mean and variance of Y are well known, so E(Y^2) can be computed using:

substitute the known mean and variance of Y:

using this with result 1, we now have:

Now compute the variance of X

As always:

Substitute known values:

so the standard deviation (sd) satisfies:

Re-arrange for :

substituting numbers in gives a parameter value of .364 if the standard deviation is 38.8, as per your example.

Post if you need a bit more explanation. If you are not comfortable using symmettry or referring to the properties of variable Y, it should be possible to do these integrals directly (but you will have to break them up, treating positive and negative values of X seperately).