91Ó°ÊÓ

The Cumulative Distribution Function, or CDF, is a pivotal concept when dealing with the exponential distribution. The CDF of a random variable provides the probability that the variable takes a value less than or equal to a particular value.
For the exponential distribution, this is formulated as \( F(t) = 1 - e^{-t/\beta} \), where \( \beta \) is the mean of the distribution and \( t \) is the time. This formula allows us to calculate the probability up to a certain time \( t \).
In our exercise, the manager's interviews are modeled with an exponential distribution with \( \beta = 2 \) hours. This means the average time for an interview is 2 hours, and we want to calculate the probability an interview finishes within 15 minutes (0.25 hours).
Calculating \( F(0.25) \) lets us understand the likelihood of the manager completing the interview before the next applicant arrives, which is crucial for scheduling and analyzing the waiting times.

Calculating the probability in an exponential distribution involves using both the CDF and recognizing its complementary nature.
For any given time \( t \), the probability that our event, such as the manager finishing an interview, takes longer than \( t \) follows as \( P(T > t) = 1 - F(t) \).
Here, \( T \) is the duration of the interview. To solve our exercise, which seeks the probability that an interview lasts more than 15 minutes, we must calculate \( 1 - F(0.25) \).
From the CDF expression, this gives us \( P(T > 0.25) = e^{-0.25 / 2} \). Simplifying further, \( e^{-0.125} \) is approximately 0.882.
This calculation provides us with the probability that an applicant arriving at 8:15 AM will need to wait because the manager is still busy interviewing the previous applicant.

The waiting time problem typically involves determining how long one might have to wait, given a random process such as an interview schedule.
In our example, this boils down to calculating the likelihood that a second scheduled applicant waits for the manager to finish an ongoing task.
In real-life scenarios, reducing waiting times is crucial for efficiency, minimizing downtime, and ensuring smooth operations.
Using the exponential distribution allows us to model such scenarios accurately, providing insights into potential scheduling improvements or changes in pacing to reduce wait times.
Once we know the probability that the first interview exceeds 15 minutes is about 0.882, the next step might involve policy changes like longer time slots or breaks to avert overlap and minimize waiting times.