The times in minutes needed to collect the tolls from motorists crossing a toll bridge has the probability density function $f(x)=2\exp(-2x), x\in[0,\infty)$
A motorist approaches the bridge and counts 50 vehicles waiting in a queue to pay the toll. Only one toll booth is in operation. Use the central limit theorem to find the approximate probability that a motorist will have to wait more than 25minutes before reaching the front of the queue
Any ideas how to start this
Hint: The exponential with density function $\lambda e^{-\lambda x}$ ($x\gt 0$) has mean $\frac{1}{\lambda}$ and variance $\frac{1}{\lambda^2}$.
Let $X_1$ the the service time of the first person in line, $X_2$ be the service time of the second, and so on up to $X_{50}$.
Our waiting time $W$ is given by the sum $$W=X_1+X_2+\cdots +X_{50}.$$ The $X_i$ are by assumption identically distributed. Let us assume they are independent.