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The probability density function, or PDF, is a fundamental concept in probability theory. It provides the likelihood of a random variable falling within a particular range of values. For the exponential distribution, which is often used to model time until an event occurs, the PDF has an exponential form. In this case:\[ f(x) = 0.2 e^{-2|x|} \]where the factor \(0.2\) ensures the function integrates to 1, making it a valid PDF. The presence of \(|x|\) indicates that the distribution is symmetric about the origin, addressing both positive and negative values.To ensure that a given PDF is legitimate, it must adhere to two principles:

• Non-negativity: \( f(x) \geq 0 \) for all \( x \).
• Normalization: the integral of \( f(x) \) over its entire range is 1.

For the exponential PDF discussed here, the integral from \(-\infty\) to \(\infty\) confirms that it adds up to 1. Thus, it meets the criteria of a PDF, representing a complete set of probabilities.

The cumulative distribution function, or CDF, accumulates the probabilities from the PDF to give the probability that a random variable is less than or equal to a certain value. For an exponential distribution, the CDF forms a continuous curve that rises smoothly from 0 to 1.For this specific example where \( f(x) = 0.2 e^{-2|x|} \), the CDF is defined piecewise, due to the absolute value in the PDF:

• For \( x < 0 \): \\[ F(x) = 0.5 - 0.1 e^{2x} \]
• For \( x \geq 0 \): \\[ F(x) = 0.5 + 0.1 (1 - e^{-2x}) \]

These expressions smoothly transition at \( x = 0 \), where \( F(0) = 0.5 \). The CDF starts at 0 as \( x \to -\infty \) and approaches 1 as \( x \to \infty \), representing all cumulative probabilities of the range.

Calculating probabilities using the CDF involves inserting specific values into the CDF and evaluating the resultant expressions. Each expression in the CDF represents probability accumulation up to that point.1. **Probability \(X \leq 0\):** This is directly given by the CDF at zero, \[ F(0) = 0.5 \], indicating that there is a 50% chance of obtaining a value less than or equal to zero.2. **Probability \(X \leq 2\):** Using the CDF expression for \(x \geq 0\), \[ P(X \leq 2) = 0.5 + 0.1 (1 - e^{-4}) \approx 0.5933 \].3. **Probability \(-1 \leq X \leq 2\):** This is determined by subtracting the CDF values, \[ P(-1 \leq X \leq 2) = F(2) - F(-1) = 0.05(-e^{-4} + e^{-2}) \approx 0.0821 \].4. **Probability \(X > 2\):** Complement of \(P(X \leq 2)\), calculated as \[ P(X > 2) = 1 - 0.5933 = 0.4067 \].These calculations illustrate how CDF values yield specific probabilities useful for practical interpretation in real-world scenarios.