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The probability density function (PDF) is crucial in understanding how probabilities are distributed across different outcomes for a random variable. In this exercise, the given PDF is \( f(x) = 0.1 e^{-0.2|x|} \). It is specific to an exponential distribution, known for decreasing exponentially as the variable moves away from zero in both directions - positive and negative. - The PDF gives the probability of the variable falling within a particular range of values.- The term \( e^{-0.2|x|} \) represents the exponential decay, where the absolute value \(|x|\) ensures symmetry around the y-axis.A legitimate PDF needs to satisfy two criteria:

• It must be non-negative for all possible values of the random variable.
• The integral over the entire range of the random variable must equal 1, ensuring total certainty that the variable takes some value.

The task verifies legitimacy by successfully integrating the function to 1. This confirms that our PDF is a valid representation of a probability distribution over the continuous random variable \(X\).

The cumulative distribution function (CDF) is another essential concept that describes the probability that a random variable is less than or equal to a certain value. If you think of the PDF as a point-specific measure, the CDF offers a cumulative perspective.For the exponential distribution, the CDF is calculated by integrating the PDF from \(-\infty\) to a specific value \(x\). For our variable \(X\):- For \(x \geq 0\), the CDF is \( F(x) = 0.5 + 0.5(1 - e^{-0.2x}) \).- For \(x < 0\), it's \( F(x) = 0.5(1 - e^{0.2x}) \).The behavior of the CDF reflects the distribution of probabilities over the range of \(X\). It starts at 0 and smoothly increases to reach 1 as \(x\) goes to infinity. Observing the CDF graph provides insight into the shifting probability as \(X\) increases or decreases.

Probability calculations based on the CDF are fundamental for interpreting the likelihood of different outcomes. In this exercise, several probabilities were computed using the CDF.- \(P(X \leq 0) = F(0)\) which is 0.5. This reflects an equal likelihood for positive and negative errors, given the symmetry.- \(P(X \leq 2)\), calculated from the CDF for \(x = 2\), equals about 0.665, showing a significant probability of errors being less than 2 nautical miles.- For \(P(-1 \leq X \leq 2)\), the probability is determined from the difference in the CDF values at 2 and -1, resulting in 0.255.These exercises reinforce the use of the CDF in computing the probability of a variable falling within certain intervals, crucial in navigation error analysis or any application involving probability distributions.

Probability integration involves calculating the probability over a range by integrating the PDF between specific limits. This is a key operation in understanding the behavior of continuous random variables.In the solution, splitting the integration to calculate the total probability is essential. The symmetry of the function around zero facilitates easier computations, allowing you to focus on one side and then double the result:\[\int_{-\infty}^{\infty} 0.1 e^{-0.2 |x|} \ dx = 2 \times \int_{0}^{\infty} 0.1 e^{-0.2 x} \ dx\]This method confirms that the entire area under the PDF curve equals 1, validating it as a probability distribution. Mastery of probability integration is essential for deeper statistical analysis, ensuring accurate calculations of probabilities over any given range.