91Ó°ÊÓ

The concept of a probability density function, often abbreviated as PDF, is essential to understanding continuous probability distributions. For an exponential distribution like in this exercise, the PDF describes the likelihood of a particular event occurring at a specific point in time. Here, the PDF is given by the function \( f(x) = \frac{e^{-x / 10}}{10} \). This indicates that the variable \( x \) represents time in minutes after 8 a.m., and the parameter \( \lambda = \frac{1}{10} \) governs the exponential nature of the distribution.

In exponential distributions, the parameter \( \lambda \) determines the rate at which events occur. A larger \( \lambda \) implies more frequent events. The probability density function itself does not give us probabilities directly, but it helps us to understand how probabilities are determined over continuous intervals of time.

In practice, we use the PDF to derive another important function called the cumulative distribution function (CDF), which is crucial for calculating the probabilities over intervals.

To understand or calculate probabilities for a range of values in a continuous distribution, like the arrival times in this exercise, we use the cumulative distribution function (CDF). The CDF provides the probability that our random variable \( x \) will take a value less than or equal to a certain value. For the exponential distribution defined here, the CDF is:

\[ F(x) = 1 - e^{-x/10} \]

This function emerges by integrating the probability density function over the range of interest. It simplifies the process of finding probabilities for intervals and is particularly useful for computations. For instance, to find out when the first customer arrives before 9:00 A.M., which is 60 minutes after 8:00 A.M., we plug in \( x = 60 \) in the CDF as shown:

• \( F(60) = 1 - e^{-60/10} = 1 - e^{-6} \approx 0.9975 \)

So, there's a 99.75% chance of a customer's arrival before 9:00 A.M.

Probability calculations is using the CDF to calculate the likelihood of certain events. In this scenario, we aim to compute probabilities between time intervals. To find the probability that a customer arrives between 8:15 and 8:30 A.M. (15 to 30 minutes after 8:00 A.M.), we can use:

• \( P(15 < X < 30) = F(30) - F(15) \)

Where,

• \( F(30) = 1 - e^{-3} \)
• \( F(15) = 1 - e^{-1.5} \)

The probability that a customer's arrival falls within this interval is then:

• \( P(15 < X < 30) = e^{-1.5} - e^{-3} \approx 0.2231 \)

Understanding how to perform probability calculation using combined CDFs aids in determining the chance of independent arrivals, such as calculating the probability that two or more customers arrive before 8:40 A.M. from a group of five. This step often requires an understanding of the binomial distribution, where cases like 2 or more out of 5 are considered by summing respective probabilities that result from CDF estimations for each scenario.