Question

Express gamma distribution in terms of sum of independent exponential random variable

Answer 1

I have a sequence T1,T2,…T1,T2,… of independent exponential random variables with paramter λλ. I take the sum S=∑ni=1TiS=∑i=1nTi and now I would like to calculate the probability density function.

Well, I know that P(Ti>t)=e−λtP(Ti>t)=e−λt and therefore fTi(t)=λe−λtfTi(t)=λe−λt so I need to find P(T1+⋯+Tn>t)P(T1+⋯+Tn>t) and take the derivative.