A student arriving at the bank has to visit two counters for service, in sequence. The service time at the first and second counter are exponentially distributed with respective average service times of 10 and 25 minutes. What is the probability that the student's service time will be greater than 30 minutes?If the average service time at both counters is 15 minutes, what is the probability that the student's service time will be greater than 30 minutes?
I'm having a hard time solving this, I know how to solve if there was one counter but if there are n counters(2 in this case) I m not able to understand what to do? I was reading about convolution of exponential random variables but didn't understand how to connect that with this problem.
The total waiting time is the sum of waiting times at each counter. You can compute CDF of the sum of 2 independent random variables $Z=X_1+X_2$ as follows: $$F_Z(z)= P(X_1+X_2 \leq z) = \int_0^z P(X_1+X_2\leq z|X_2=x)f_2(x)dx=\int_0^zP(X_1\leq z-x)f_2(x)dx= \int_0^zF_1(z-x)f_2(x)dx$$
The final probability is $$P= 1-F_z(30)$$