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The Cumulative Distribution Function (CDF) and its Properties

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The Cumulative Distribution Function (CDF) is a statistical tool that represents the probability of a random variable being less than or equal to a value. It is applicable to both discrete and continuous variables, providing a unified view of probability distributions. The CDF is non-decreasing, bounded by one, and right-continuous. Understanding the CDF is crucial for interpreting probabilities and is closely related to the Probability Density Function (PDF) through integration and differentiation.

Exploring the Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) is an essential concept in the field of statistics, representing the probability that a random variable \(X\) will assume a value less than or equal to a particular number \(x\). The CDF, symbolized by \(F(x)\), is mathematically defined as \( F(x) = P(X \le x)\). This function is applicable to both discrete and continuous random variables, offering a complete picture of the distribution of probabilities for all potential outcomes. As a fundamental tool in statistical analysis, the CDF provides a comprehensive view of a variable's probability distribution through a single, unified function.
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Relationship Between Probability Density Function (PDF) and CDF

The connection between the Probability Density Function (PDF) and the CDF is crucial for continuous random variables. The PDF, denoted as \(f_X(x)\), specifies the relative likelihood of the random variable \(X\) taking on a precise value. It is characterized by two main conditions: the function must be non-negative (\(f_X(x) \ge 0\) for all \(x\)) and the integral of \(f_X(x)\) over the entire range of \(X\) must be equal to \(1\). The CDF is obtained by integrating the PDF from negative infinity up to the value \(x\), expressed mathematically as \( F(x) = \int_{-\infty}^x f_X(t) \, \mathrm{d} t \). Conversely, the derivative of the CDF with respect to \(x\) yields the PDF. This integral-derivative relationship is fundamental to understanding the link between these two statistical functions.

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