Tough Question on Exponential Distribution - For Experts

Hi,
I have a random variable (r.v. A), which has an exponential distribution -
A ~ exp(lambda1);

I have another random variable (r.v. B), which has an exponential
distribution as follows -
B ~ exp(A); (A being the random variable from above);

Question: What kind of density function and distribution function does B
have??


Working:

random variable A has a standard exponential distribution function:
f(A) = (lambda1)x(exp((-lambda1)x(a))) for a>0

but random variable B has a rate parameter that is non-constant!!
???

The second one looks more like the Gumbel extreme value distribution.