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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.

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## Definition and Purpose of the CDF

### CDF for discrete and continuous random variables

The CDF, symbolized by \(F(x)\), represents the probability that a random variable \(X\) will assume a value less than or equal to a particular number \(x\)

### Relationship between CDF and Probability Density Function (PDF)

Definition and properties of PDF

The PDF, denoted as \(f_X(x)\), specifies the relative likelihood of the random variable \(X\) taking on a precise value and must be non-negative and integrate to one

Integral-derivative relationship between CDF and PDF

The CDF is obtained by integrating the PDF and the derivative of the CDF yields the PDF

### Characteristics of the CDF

The CDF is non-decreasing, bounded above by one, and right-continuous, representing the cumulative probability up to a given point

## Applications of the CDF

### Use of CDF in calculating probabilities

The CDF graphically represents the area under the PDF curve up to a point \(x\), allowing for the direct calculation of probabilities

### CDF of the normal distribution

The CDF of the normal distribution, a commonly used model for random variables, is a sigmoidal curve that visually depicts the probability of a random variable falling within a specific interval

### Criteria for a valid CDF and PDF

A CDF must be non-decreasing and cannot exceed one, while a PDF must be non-negative and integrate to one, in order to be considered legitimate