91Ó°ÊÓ

When dealing with a Poisson process, the time between events follows an exponential distribution. This type of distribution is perfect for modeling the waiting time until the first occurrence of an event. The rate at which events happen in a Poisson process is represented by the parameter \(\lambda\). In our exercise, \(\lambda\) is 10 hits per minute.

The waiting time in a Poisson process is the time we wait until the first event occurs. This waiting time is exponentially distributed. Imagine standing at the bus stop and wondering how long until the next bus arrives. If the buses follow a Poisson process, then the wait time between buses follows an exponential distribution. In the given problem, we utilize the exponential distribution to determine the waiting time before receiving the first website hit.

The cumulative distribution function (CDF) is essential in determining probabilities for continuous random variables like the waiting time in an exponential distribution. Specifically, the CDF of an exponential distribution gives us the probability that the waiting time \(T\) will be less than or equal to some value \(t\). For the exponential distribution with parameter \(\lambda\), the CDF is given by \[ P(T \leq t) = 1 - e^{-\lambda t} \]. In our problem, we set this formula equal to 0.95 and solve for \(t\) to find out how long we should expect to wait to achieve a 95% chance of at least one hit to the website. Solving the equation \[ 1 - e^{-10t} = 0.95 \], we find \(-ln(0.05)/10\), approximately 0.2996 minutes. This calculation gives us the 95th percentile for the waiting time distribution.