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The probability density function (PDF) is a fundamental concept in statistics used to specify the probability of a random variable falling within a particular range of values. For continuous random variables, the PDF provides the height of the curve at any given point along the distribution. This specific curve does not give probabilities directly, but rather represents the entire probability distribution.

In the context of the exponential distribution, the PDF is particularly useful as it describes the likelihood of random events and can model time until an event occurs. For this exercise, the PDF is given by:

• \( f(x) = 0.1 e^{-0.2|x|} \) where \(-\infty < x < \infty\).

**Key Characteristics**:

• The PDF is non-negative for all values of \(x\).
• The total area under the curve of the PDF is 1, confirming it represents a legitimate probability distribution.
• This specific PDF is modified by the absolute value notation \(|x|\), making it suitable to accommodate both negative and positive errors.

To check if a function is a proper PDF, you integrate it over its entire range. If the result is 1, it is confirmed to represent a valid probability distribution.

The cumulative distribution function (CDF) is another fundamental tool in probability and statistics. It calculates the probability that a random variable takes on a value less than or equal to a certain amount. For continuous random variables, the CDF is derived by integrating the PDF from the start of the distribution up to point \(x\).

In our exercise, we are dealing with a distribution where the CDF can be expressed as:

• For \(x \geq 0:\) \(P(X \leq x) = 0.5 + 0.5(1-e^{-0.2x})\)
• For \(x < 0:\) \(P(X \leq x) = \frac{1}{2}e^{0.2x}\)

**Properties of the CDF include**:

• Starts from 0 and increases to 1 as \(x\) moves from \(-\infty\) to \(+\infty\).
• It is a non-decreasing function, which means once it reaches a certain probability, it cannot go down.
• The graph is smooth and reflects the symmetric nature of the errors about the origin, with a value of 0.5 at x=0.

To visualize the CDF, imagine a curve starting from 0 on the left and smoothly rising to 1 towards the right. At \(x=0\), the CDF is exactly 0.5, and it gradually approaches 1 as \(x\) approaches positive infinity.

Calculating probabilities using the PDF and CDF allows us to understand the likelihood of different scenarios occurring within the context of the distribution. In the context of this specific exercise, different probability queries are solved using the CDF.

Let's break down some common calculations involved:

• **\(P(X \leq 0)\):** This represents the probability that the random variable \(X\) is zero or negative. It is equal to the CDF at zero, which is 0.5.
• **\(P(X \leq 2)\):** To find this probability, use the CDF formula for non-negative X. The calculation results in a probability of roughly 0.6636.
• **\(P(-1 \leq X \leq 2)\):** This involves calculating the difference of probabilities. Use the CDF values: \(P(X \leq 2)\) minus \(P(X \leq -1)\), resulting in approximately 0.4622.
• **Probability of an Error Greater Than 2 miles:** Use the complement rule. The calculation \(1 - P(X \leq 2)\) gives a probability of about 0.3364.

The above procedures highlight how the CDF aids in determining probabilities over specified intervals, leveraging the properties of the exponential distribution.