91Ó°ÊÓ

The Cumulative Distribution Function (CDF) of a continuous random variable is a fundamental concept in probability and statistics. It describes the probability that a random variable is less than or equal to a certain value. For a variable \(X\), the CDF, denoted as \(F(x)\), is defined as:

• \(F(x) = P(X \leq x)\)

The CDF is a non-decreasing function that starts at 0 and approaches 1 as \(x\) increases. It provides a complete picture of the distribution of a continuous random variable, mapping each possible value to a probability. In the exercise above, the CDF is given by different expressions based on the value of \(x\):

• \(F(x) = 0\) for \(x \leq 0\)
• \(F(x) = \frac{x}{4}\left[1+\ln \left(\frac{4}{x}\right)\right]\) for \(0 < x \leq 4\)
• \(F(x) = 1\) for \(x > 4\)

These expressions indicate the behavior of the random variable across its possible range, illustrating how the probability is distributed.

For continuous random variables, the Probability Density Function (PDF) gives a functional form that influences the likelihood of a variable taking on a particular value. However, the probability of \(X\) being exactly a single value \(x\) is technically zero. Instead, the PDF, often denoted \(f(x)\), describes the density of the probability around \(x\). Integrating the PDF over a range gives the probability that \(X\) lies within that range:

• \(P(a < X < b) = \int_{a}^{b} f(x) \, dx\)

From the exercise, the PDF can be found by differentiating the CDF over the interval where it is continuous, specifically for \(0 < x \leq 4\):

• First term: Derivative of \(\frac{x}{4}\) is \(\frac{1}{4}\).
• Second term: Product rule for \(\frac{x}{4} \ln\left(\frac{4}{x}\right)\)

After simplification, this results in:

• \(f(x) = \frac{1}{4}(1 + \ln\left(\frac{x}{4}\right))\)

This expression represents the normalized rate at which probability is assigned along the range of \(X\).

Integration plays a crucial role in statistics, particularly in handling continuous random variables. When dealing with PDFs, integration allows us to calculate probabilities over intervals. This process sums up infinitesimal slices of probability across a specified range:

• The definite integral of the PDF from \(a\) to \(b\) gives the probability that a continuous random variable falls within this interval: \(\int_{a}^{b} f(x) \, dx\).
• Integrating the PDF over its entire range gives 1, confirming that the total probability is distributed.

In the provided exercise, integration is implicitly used to transition from the PDF to probability outcomes. For example, knowing \(P(1 \leq X \leq 3)\) involves the differences \(F(3) - F(1)\), which conceptually relies on integrating the PDF across this range. Differentiating the CDF to find the PDF is the reverse procedure, using calculus to articulate how probability is distributed along a continuous spectrum. This forms the backbone of probability interpretations for continuous random variables.