I'm pretty stuck with the following question, any help would be appreciated
A system is composed of 5 subsystems, each of them has a random independent lifetime, exponentially distributed with mean = 1.
The system works only if all the subsystems are working.
What is the variance of the whole system's lifetime?
well... I calculated it now.
if X~exp(1) and Y~exp(1) and Z = min(X,Y)
then the probability density function of Z is 2exp(-2z) for z>0 and 0 otherwise.
So it's also exponential with parameter=2.
does that mean that the minimum of the five would be also exponential with a parameter of 5? and then the answer should be 1/25..
but as far as I recall, the sum of two exponentials is an exponential with a parameter that equals the sum of parameters, is it true? does it mean that minimum and sum of exponentially distributed random variables are equivalent?...
Your computation is right. More generally, if and are exponential r.v. of parameters and , then is an exponential r.v. with parameter . If you apply this 4 times (first, the min of the lifetimes of the first two subsystems, then the minimum of the first two and the third,...), you get exactly what you guessed: the lifetime is an exponential r.v. of parameter 5.
As for the sum, be careful, it is called a gamma distribution, and not exponential.