Cumulative Distribution Function

A **cumulative distribution function** (CDF) is a fundamental concept in probability theory and statistics that describes the probability that a random variable takes on a value less than or equal to a specific value. The CDF provides a complete description of the probability distribution of a real-valued random variable.

Definition

Formally, the CDF of a random variable \(X\) is defined as: \[ F_X(x) = P(X \leq x) \] where \(P\) denotes the probability. For every real number \(x\), \(F_X(x)\) represents the probability that the random variable \(X\) will take a value less than or equal to \(x\).

Properties

The CDF has several important properties:

**Non-decreasing**: The CDF is a non-decreasing function. If \(a \leq b\), then \(F_X(a) \leq F_X(b)\).
**Right-continuous**: The CDF is right-continuous, meaning that \(\lim_{x \to c^+} F_X(x) = F_X(c)\) for any real number \(c\).
**Limits**: The limits of the CDF as \(x\) approaches \(-\infty\) and \(+\infty\) are:

 \[ \lim_{x \to -\infty} F_X(x) = 0 \]
 \[ \lim_{x \to +\infty} F_X(x) = 1 \]

Types of Cumulative Distribution Functions

CDFs can be classified based on the type of random variable they describe:

Discrete CDF

For a discrete random variable, the CDF is a step function. Suppose \(X\) is a discrete random variable taking values \(x_1, x_2, \ldots, x_n\) with probabilities \(p_1, p_2, \ldots, p_n\). The CDF is given by: \[ F_X(x) = \sum_{x_i \leq x} p_i \]

Continuous CDF

For a continuous random variable, the CDF is a continuous function. If \(X\) has a probability density function (PDF) \(f_X(x)\), the CDF is obtained by integrating the PDF: \[ F_X(x) = \int_{-\infty}^x f_X(t) \, dt \]

Applications

CDFs are widely used in various fields such as statistics, finance, engineering, and science. Some of the key applications include:

**Statistical Inference**: CDFs are used to derive properties of estimators and test statistics.
**Risk Management**: In finance, CDFs help in assessing the risk of investment portfolios.
**Reliability Engineering**: CDFs are used to model the life distribution of products and systems.

Relationship with Other Functions

The CDF is closely related to other functions in probability theory:

Probability Density Function (PDF)

For continuous random variables, the PDF \(f_X(x)\) is the derivative of the CDF: \[ f_X(x) = \frac{d}{dx} F_X(x) \]

Survival Function

The survival function \(S_X(x)\) is the complement of the CDF: \[ S_X(x) = 1 - F_X(x) \] It represents the probability that the random variable \(X\) is greater than \(x\).

Quantile Function

The quantile function \(Q(p)\) is the inverse of the CDF, defined for \(0 < p < 1\) as: \[ Q(p) = \inf \{ x \in \mathbb{R} : F_X(x) \geq p \} \] It gives the value below which a given percentage of observations in a dataset fall.

Examples

Example 1: Discrete Random Variable

Consider a discrete random variable \(X\) that takes values 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3, respectively. The CDF is: \[ F_X(x) = \begin{cases} 0 & \text{if } x < 1 \\ 0.2 & \text{if } 1 \leq x < 2 \\ 0.7 & \text{if } 2 \leq x < 3 \\ 1 & \text{if } x \geq 3 \end{cases} \]

Example 2: Continuous Random Variable

Consider a continuous random variable \(X\) with a uniform distribution on the interval \([0, 1]\). The PDF is \(f_X(x) = 1\) for \(0 \leq x \leq 1\). The CDF is: \[ F_X(x) = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } 0 \leq x \leq 1 \\ 1 & \text{if } x > 1 \end{cases} \]

Mathematical Properties

Monotonicity

The CDF is a non-decreasing function, meaning that for any \(a \leq b\), \(F_X(a) \leq F_X(b)\). This property ensures that as the value of the random variable increases, the probability does not decrease.

Right-Continuity

The CDF is right-continuous, which means that for any \(x\), the limit of \(F_X(y)\) as \(y\) approaches \(x\) from the right is equal to \(F_X(x)\): \[ \lim_{y \to x^+} F_X(y) = F_X(x) \]

Limits

The limits of the CDF as \(x\) approaches \(-\infty\) and \(+\infty\) are: \[ \lim_{x \to -\infty} F_X(x) = 0 \] \[ \lim_{x \to +\infty} F_X(x) = 1 \]

Inversion

For a continuous random variable, the CDF can be inverted to find the quantile function. If \(U\) is a uniform random variable on \([0, 1]\), then \(X = F_X^{-1}(U)\) has the same distribution as \(X\).

Practical Considerations

Empirical CDF

In practice, the CDF can be estimated from data using the empirical CDF. Given a sample \(x_1, x_2, \ldots, x_n\), the empirical CDF \(F_n(x)\) is defined as: \[ F_n(x) = \frac{1}{n} \sum_{i=1}^n I(x_i \leq x) \] where \(I\) is the indicator function.

Computational Methods

Computing the CDF for complex distributions may require numerical methods. Techniques such as Monte Carlo simulation and numerical integration are often used.

Conclusion

The cumulative distribution function is a crucial tool in probability theory and statistics, providing a comprehensive description of the distribution of a random variable. Its properties and relationships with other functions make it an essential concept for understanding and analyzing probabilistic models.