# Using central limit theorem on a probability density function

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

• What exactly are you having problems with? do you know what the central limit theorem says? Do you know the mean and variance of the given pdf? Apparently we are to assume the toll collection times are IID. – Tyler Feb 25 '14 at 15:31

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.