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The cumulative distribution function (CDF) is a crucial concept in probability and statistics. It represents the probability that a random variable will take a value less than or equal to a specific value. In other words, for a duration of call, the CDF tells us the likelihood that the call lasts no longer than that amount of time.
The formula for the CDF, based on a given probability density function (PDF), is written as:

• \( F(x) = \int_{0}^{x} p(t) \, dt \)

This integral accumulates the probabilities from zero to \(x\), forming a function that starts at zero and increases to one as \(x\) increases. For the exercise given, the PDF is \( p(x) = 0.4 e^{-0.4 x} \), leading to the CDF:

• \( F(x) = 1 - e^{-0.4x} \).

This CDF indicates how likely it is for a call's duration to be less than or equal to any given time \(x\). This step is fundamental to solve further probability questions by providing cumulative probabilities over a range of times.

The exponential distribution is a probability distribution that describes time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It's often used to model the time until an event happens, such as the duration of a phone call until it ends.

In the case of this exercise, the duration of phone calls follows an exponential distribution described by the density function \( p(x) = 0.4 e^{-0.4 x} \). This type of distribution is characterized by its memoryless property, meaning that the likelihood of a future event occurring does not depend on past behavior.

Key characteristics of exponential distribution include:

• The mean, which is the average time between events. For the given PDF, this is \( \frac{1}{0.4} = 2.5 \) minutes.
• The rate parameter, often denoted by \( \lambda \), which in this context is \(0.4\) per minute.

Understanding these characteristics of the exponential distribution helps us comprehend various scenarios' probabilities and deal with problems like calculating how long a phone call might take.

Probability calculation involves determining the chance that a certain event will occur, often expressed as a percentage. To solve problems using the exponential distribution, like those in this exercise, you can utilize the cumulative distribution function (CDF).

Steps involve:

• Identifying the range of interest, such as calls lasting between 1 and 2 minutes.
• Calculating \( F(x) \) at the range endpoints using the CDF formula. For example, \( F(2) - F(1) \) provides the probability of calls between 1 to 2 minutes.
• For a range starting from zero, the end probability is simply \( F(x) \).
• To find probabilities for intervals starting above zero, like 3 minutes and more, utilize complementary probability: \( 1 - F(x) \).

For the provided solutions:

• Cumulative probability \(P(x \leq 1)\) gives a 33% chance for calls lasting 1 minute or less.
• Probability of calls lasting between 1 and 2 minutes was approximately 24.5%.
• Calculating for calls lasting 3 or more minutes yields a 30.1% probability.

By systematically using the rules of probability calculation, students can effectively analyze and solve a range of similar exercises across various applications.