2. The arrivals of customers at a drive-in banking window are governed by the Poisson distribution. Each customer requires 5 minutes to transact his or her business at the drive-in window. The average customer arrival rate is 10 per hour. A customer arrives at random. What is the probability that there will be no line and he can be served immediately? NOTE: You should understand the relationship between the Poisson and Exponential distributions to solve this problem.
To find the probability that there will be no line and the customer can be served immediately, we need to use the relationship between the Poisson and Exponential distributions.
Let's denote λ as the customer arrival rate per hour, which is given as 10.
In this case, the average time between customer arrivals (interarrival time) follows the Exponential distribution with parameter β, where β = 1/λ.
So, β = 1/10 = 0.1.
We need to find the probability that the interarrival time is less than or equal to 0. By definition, the Exponential distribution models the time until the first event occurs.
The probability that the customer can be served immediately is the probability that no time elapses between the previous customer leaving and the current customer arriving.
This can be calculated as 1 minus the probability that the interarrival time is greater than 0.
Using the Exponential distribution's cumulative distribution function (CDF), we can calculate the probability as follows:
P(T ≤ 0) = 1 - exp(-β * 0)
Here, exp is the exponential function.
Since exp(-β * 0) equals 1, the probability can be simplified as:
P(T ≤ 0) = 1 - 1 = 0
Therefore, the probability that the customer can be served immediately and there will be no line is 0.
To solve this problem, we need to understand the relationship between the Poisson and Exponential distributions. The Poisson distribution describes the number of events that occur in a fixed interval of time or space. On the other hand, the Exponential distribution models the time between events in a Poisson process.
Given that the average customer arrival rate is 10 per hour, it means that the average time between customer arrivals (inter-arrival time) is 1/10 hour or 6 minutes.
To find the probability that there will be no line and the customer can be served immediately, we need to determine the time it takes for the next customer to arrive. This time follows an Exponential distribution with a mean of 6 minutes.
The probability density function (PDF) of the exponential distribution is given by:
f(t) = λ * exp(- λt)
Where:
- f(t) is the probability density function
- λ is the rate parameter, equal to the reciprocal of the mean (λ = 1/mean)
- t is the time between events
In this case, λ = 1/6 (since the mean is 6 minutes).
To find the probability that the next customer arrives immediately (in 0 minutes), we substitute t = 0 into the exponential PDF:
f(0) = (1/6) * exp(-(1/6)*0) = 1/6
So the probability that there will be no line and the customer can be served immediately is 1/6 or approximately 0.1667.