# Derive the probability density function of $Z = T_1 + T_2$.

Let $$T_1$$ be the waiting time until the first call in a call center and let $$T_2$$ be the waiting from the first call until the second call.
Assume that both $$T_1$$ and $$T_2$$ have an exponential distribution with expectation $$0.1$$ minutes. Furthermore, $$T_1$$ and $$T2$$ are independent.
Derive the probability density function of $$Z = T_1 + T_2$$.

Since $$E[T_1]=E[T_2]=1/10$$, $$T_1$$ and $$T_2$$ have a $$Exp(10)$$ distribution, each with a probability density function of $$f(x)=10*e^{-10*x}$$, then how can I determine the probability density function of the sum?
I tried to add both but it doesn't work, can someone help me please? Thanks in advance!!

• you can try using the moment-generating functions if you know what that is. – J.F Jan 19 at 19:01
• I don't know, sorry.. – Mark Jacon Jan 19 at 19:03
• Use the basic formula giving the PDF of the sum of some independent random variables as the convolution of their respective PDFs, which is most probably in your notes, and you are done. – Did Jan 19 at 19:41