Solving the Exponential Random Variable Problem for Doctor Appointments

In the medical field, time management is of utmost importance. This is especially true when it comes to scheduling appointments for patients. Doctors must ensure that their patients receive the necessary care and attention without compromising on the time allocated for other appointments. In this post, we will look at a scenario where a doctor has scheduled two appointments, one at 1 P.M. and the other at 1:30 P.M. The amounts of time that appointments last are independent exponential random variables with mean 30 minutes.

Problem Statement:

A doctor has scheduled two appointments, one at 1 P.M. and the other at 1:30 P.M. The amounts of time that appointments last are independent exponential random variables with mean 30 minutes. Assuming that both patients are on time, find the expected amount of time that the 1:30 appointment spends at the doctor’s office.

Solution:

The expected amount of time spent at the doctor’s office for an exponential random variable with mean 30 minutes can be calculated as follows:

E(X) = 1/λ

where λ is the rate parameter and is equal to 1/mean. In this case, λ = 1/30 = 0.03333.

Therefore, the expected amount of time spent at the doctor's office for each appointment is 1/λ = 1/0.03333 = 30 minutes.

The expected amount of time that the 1:30 appointment spends at the doctor’s office is also 30 minutes. This is because the amount of time that appointments last is an independent exponential random variable with a mean of 30 minutes. The expected value for each appointment remains the same regardless of the appointment time. By understanding the concept of exponential random variables, doctors can better manage their appointments and provide the necessary care and attention to each patient within the allotted time.