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The Probability Density Function (PDF) of an exponential distribution describes how probabilities are spread across different values. In our case, the PDF is given by the formula: \[f(x) = \frac{1}{3} e^{-x / 3}\] This function helps us understand the likelihood of different outcomes for the random variable \(x\), where \(x\) can be any non-negative number. The PDF increases or decreases exponentially, depending on the value of \(x\).
The most notable feature is the presence of \(e^{-x/3}\), an exponential term that decreases as \(x\) increases. The coefficient \(\frac{1}{3}\) means that the average rate at which events happen per unit time is constant. As \(x\) increases, the probability density decreases, indicating that very high values of \(x\) are less likely.

The Cumulative Distribution Function (CDF) of an exponential distribution summarizes the probability that a random variable \(x\) is less than or equal to a specific value. For our situation, the CDF can be expressed as:\[F(x) = 1 - e^{-x/3}\]This function is derived from the PDF by integrating the PDF from \(0\) to \(x\). The CDF is especially useful for calculating probabilities over a range of values.
It provides an accumulated total of probabilities, up to a certain point \(x\). By plugging different values of \(x\) into the CDF, you can find the probability for "up to" situations, such as \(P(x \leq x_0)\). For example, in this exercise, to find \(P(x \leq 2)\), just substitute \(x = 2\) in the CDF to find the result.

The Complement Rule is a helpful concept that allows us to calculate probabilities when direct computation is cumbersome. It states that the probability of an event happening is equal to one minus the probability of the event not happening. Mathematically, this is expressed as:\[P(A) = 1 - P(A^c)\]In this exercise, we used the Complement Rule to compute \(P(x \geq 3)\) by calculating \(P(x < 3)\) through the CDF first. Once we have \(P(x < 3)\), applying the complement, we get \(P(x \geq 3) = 1 - P(x < 3)\).
This shortcut makes it more convenient compared to finding probabilities directly, especially when dealing with inequalities like \(P(x \geq x_0)\). This technique becomes incredibly useful in situations where the CDF gives direct calculations for only up to certain values.

Once we have the CDF and utilization of complement rule, we proceed to the actual probability calculations. We calculated different probabilities in the step-by-step solution using the CDF and Complement Rule, including \(P(x \leq 2)\), \(P(x \leq 5)\), and \(P(2 \leq x \leq 5)\).- To find \(P(x \leq 2)\), we simply plug \(x = 2\) into the CDF: \(F(2) = 1 - e^{-2/3}\).- For \(P(x \leq 5)\), it’s the same procedure with \(F(5) = 1 - e^{-5/3}\).- For \(P(2 \leq x \leq 5)\), which is a probability over a range, we calculate it as \(F(5) - F(2)\). This method combines the probabilities obtained in the previous steps to find the desired value.
By understanding and using both the CDF and Complement Rule, you have a powerful toolkit for calculating different probabilities associated with the exponential distribution.