Question

Problem 10. Times between accidents are identically independently distributed exponential with mean 2. The fourth accident triggers an alarm. Find the density function of the time of this alarm.

Answer 1

Let X1, X2, X3, X4 be the times between accidents until the fourth accident

Let T be the time of the alarm. Then T = X1 + X2 + X3 + X4

Given, X_i ~ Exp() where   = 1/mean = 1/2

the moment generating function is,

The moment generating function of T = X1 + X2 + X3 + X4 is,

which is moment generating function of Gamma distribution with Shape, k = 4 and Scale = 2.

Thus, the density function of the time of this alarm is Gamma(k = 4, = 2)

for t > 0