Cumulative Distribution Functions

Cumulative Distribution Functions (CDFs) are key in probability theory, showing the likelihood that a random variable is less than or equal to a certain value. They help describe distributions, calculate probabilities, and connect to other concepts like Probability Density Functions (PDFs).

1. Definition of Cumulative Distribution Function (CDF)

   • A CDF, denoted as F(x), represents the probability that a random variable X takes on a value less than or equal to x.
   • It is a non-decreasing function that ranges from 0 to 1.
   • The CDF provides a complete description of the probability distribution of a random variable.

2. Properties of CDFs

   • F(x) is right-continuous, meaning it approaches the limit from the right.
   • The limits are: F(-∞) = 0 and F(∞) = 1.
   • CDFs are non-decreasing; if x1 < x2, then F(x1) ≤ F(x2).

3. Relationship between CDF and Probability Density Function (PDF)

   • The PDF, denoted as f(x), is the derivative of the CDF: f(x) = dF(x)/dx for continuous variables.
   • The area under the PDF curve between two points gives the probability of the random variable falling within that range.
   • For discrete variables, the PDF is represented as a probability mass function (PMF), and the CDF is the sum of probabilities up to x.

4. Continuous vs. Discrete CDFs

   • Continuous CDFs are derived from continuous random variables and are smooth functions.
   • Discrete CDFs consist of step functions, reflecting the probabilities of specific outcomes.
   • Both types of CDFs fulfill the same fundamental role in describing the distribution of random variables.

5. Calculating probabilities using CDFs

   • To find the probability that a random variable falls within a specific range, use: P(a < X ≤ b) = F(b) - F(a).
   • For discrete variables, probabilities can be calculated by summing the differences in the CDF values.
   • CDFs simplify the process of finding cumulative probabilities for complex distributions.

6. Inverse CDF and its applications

   • The inverse CDF, or quantile function, provides the value of x for a given probability p, denoted as F⁻¹(p).
   • It is used in simulations and generating random samples from a specified distribution.
   • The inverse CDF is crucial in statistical methods, such as hypothesis testing and confidence interval estimation.

7. Joint CDFs for multiple random variables

   • A joint CDF describes the probability that two or more random variables simultaneously fall within specified ranges.
   • It is denoted as F(x, y) for two variables and can be extended to more variables.
   • Joint CDFs are essential for understanding the relationships and dependencies between multiple random variables.

8. CDFs for common probability distributions (e.g., Normal, Exponential, Uniform)

   • Each probability distribution has a specific CDF that characterizes its behavior.
   • The Normal distribution has a bell-shaped CDF, while the Exponential distribution has a rapidly increasing CDF.
   • Understanding these CDFs helps in applying statistical methods and making predictions based on the distribution type.

9. Empirical CDF and its use in data analysis

   • The empirical CDF is constructed from observed data and estimates the true CDF of a population.
   • It provides a non-parametric way to visualize and analyze data distributions.
   • Empirical CDFs are useful for comparing sample distributions and assessing goodness-of-fit for theoretical models.

10. CDF transformations and their applications

    • CDF transformations involve applying functions to the CDF to derive new distributions or to standardize data.
    • Common transformations include the Box-Cox transformation and the logit transformation.
    • These transformations are used to meet statistical assumptions, improve model fit, and facilitate analysis.