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The Probability Density Function (PDF) is a key concept within the study of continuous probability distributions. It describes the likelihood of a random variable taking on a specific value. For an exponential distribution, the PDF is often expressed in the form:

\[ f(x) = \frac{1}{\lambda} e^{-x/\lambda} \]Here, \( \lambda \) represents the rate parameter, which determines the "speed" at which events occur. In our specific problem, the PDF is given as \( f(x) = \frac{1}{3} e^{-x/3} \), implying a value of \( \lambda = 3 \). This means events occur at a certain average rate over time, shaping the behavior of the distribution. Understanding the PDF is crucial because it lays the groundwork for further probability calculations and developing other functions like the Cumulative Distribution Function (CDF).

The Cumulative Distribution Function (CDF) provides us with the probability that a random variable is less than or equal to a particular value. It's the sum of probabilities from the start of the event until a particular value. For an exponential distribution with PDF \( f(x) = \frac{1}{\lambda} e^{-x/\lambda} \), the CDF is expressed as:

\[ F(x) = 1 - e^{-x/\lambda} \]In the context of our problem, given that \( \lambda = 3 \), the CDF becomes:

\[ F(x) = 1 - e^{-x/3} \]This CDF helps us calculate probabilities for problems like the ones presented in the exercise, allowing for the determination of the likelihood of events within specified ranges. The transformation from PDF to CDF forms the backbone of many calculations in probability theory.

Calculating probabilities using exponential distribution involves applying the CDF. Below are the key calculations using the CDF of our problem:

• To find \( P(x \leq x_0) \), use the formula: \( P(x \leq x_0) = 1 - e^{-x_0/3} \).
• To calculate \( P(x \leq 2) \): substitute \( x_0 = 2 \) into the formula: \( P(x \leq 2) = 1 - e^{-2/3} \approx 0.4866 \).
• For \( P(x \geq 3) \), use: \( P(x \geq 3) = e^{-1} \approx 0.3679 \).
• Determine \( P(x \leq 5) \) by substituting \( x_0 = 5 \): \( P(x \leq 5) = 1 - e^{-5/3} \approx 0.8111 \).
• For a range such as \( P(2 \leq x \leq 5) \), subtract to find interval probabilities: \( P(2 \leq x \leq 5) = 0.8111 - 0.4866 = 0.3245 \).

These steps illustrate the practical use of CDF in solving probability questions effectively.

The exponential probability distribution is a crucial tool for modeling events that occur continuously and independently at a constant average rate, such as time between arrivals in a queue. It is characterized by a memoryless property, meaning that the probability of future events does not depend on the past, only on the present.

• The exponential distribution is defined by its rate parameter \( \lambda \), where a smaller \( \lambda \) indicates events occur more slowly.
• It is widely used in various fields such as physics, finance, and queueing theory due to its simplicity and ability to model real-world scenarios accurately.
• Understanding the exponential distribution goes beyond solving exercises; it helps in predicting behaviors in systems and processes across disciplines.

By grasping the exponential probability distribution, students gain valuable insight into complex systems and the ability to make informed predictions based on this mathematical model.