Cumulative distribution functions (CDFs) are essential tools in probability theory. They give us a complete picture of a random variable's behavior, showing the likelihood of it falling below any given value. CDFs are versatile, applicable to both discrete and continuous variables.

CDFs connect to the broader study of continuous random variables by providing a unified framework for analysis. They allow us to calculate probabilities, find percentiles, and generate random samples. Understanding CDFs is crucial for grasping more advanced concepts in probability and statistics.

Cumulative Distribution Functions

Definition and Basic Properties

• Cumulative distribution function (CDF) F(x) for random variable X represents probability X takes value less than or equal to x $F(x) = P(X ≤ x)$
• Non-decreasing function $F(x₁) ≤ F(x₂)$ for all $x₁ < x₂$
• Approaches 0 as x approaches negative infinity and 1 as x approaches positive infinity
• Continuous function for continuous random variables
• Right-continuous, meaning $\lim_{x \to x₀⁺} F(x) = F(x₀)$
• Step function with jumps at possible values of X for discrete random variables
• Probability of X taking specific value a given by $P(X = a) = F(a) - \lim_{x \to a⁻} F(x)$

Advanced Properties and Applications

• Used to generate random samples from given distribution through inverse CDF method
• Derives statistical measures (expected values, variances) through integration techniques
• Analyzes transformations of random variables, particularly for monotonic functions
• Determines stochastic dominance between different random variables or distributions
• Represents failure distribution in reliability analysis, with complement (1 - F(x)) as reliability function
• Utilized in hypothesis testing (Kolmogorov-Smirnov tests) to compare sample distributions to theoretical distributions
• Analyzes relationships between multiple random variables using joint CDFs and marginal CDFs in multivariate settings

CDFs vs PDFs

Relationship and Conversion

• Probability density function (PDF) f(x) represents derivative of CDF F(x) for continuous random variables $f(x) = \frac{dF(x)}{dx}$
• CDF obtained by integrating PDF $F(x) = \int_{-\infty}^x f(t)dt$
• Area under PDF curve between points a and b represents probability $P(a < X ≤ b) = F(b) - F(a)$
• For discrete random variables, probability mass function (PMF) replaces PDF, CDF becomes sum of PMF values up to and including x
• PDF must be non-negative and integrate to 1 over entire domain, corresponding to CDF ranging from 0 to 1
• Discontinuities in CDF correspond to point masses in probability distribution, represented by delta functions in PDF for continuous random variables

Comparative Analysis

• CDF provides cumulative probabilities, while PDF gives probability density at specific points
• CDF always exists for all random variables, while PDF exists only for continuous random variables
• CDF ranges from 0 to 1, while PDF can take any non-negative value
• CDF used directly for calculating probabilities, while PDF requires integration
• CDF always continuous from the right, while PDF may have discontinuities
• CDF monotonically increasing, while PDF can increase or decrease
• CDF used in stochastic ordering and reliability analysis, while PDF used in maximum likelihood estimation and information theory

Probabilities with CDFs

Basic Probability Calculations

• Probability of X being less than or equal to specific value a given directly by CDF $P(X ≤ a) = F(a)$
• Probability of X being greater than specific value a calculated as $P(X > a) = 1 - F(a)$
• Probability of X falling within interval [a, b] calculated as $P(a ≤ X ≤ b) = F(b) - F(a)$
• For continuous random variables, $P(X = a) = 0$, so $P(a < X ≤ b) = P(a ≤ X ≤ b) = F(b) - F(a)$
• Median of distribution found by solving $F(x) = 0.5$
• Percentiles of distribution calculated by finding inverse of CDF (quantile function)
• For discrete random variables, probabilities of specific values found by subtracting consecutive CDF values $P(X = x) = F(x) - F(x-)$

Advanced Probability Applications

• Calculating joint probabilities for multiple random variables using multivariate CDFs
• Determining conditional probabilities using CDFs of marginal and joint distributions
• Computing probabilities for functions of random variables through CDF transformations
• Analyzing order statistics using CDFs (finding distribution of maximum or minimum of sample)
• Calculating probabilities in reliability engineering (system failure probabilities)
• Determining probabilities in financial risk management (Value at Risk calculations)
• Computing probabilities in queueing theory (waiting time distributions)

Continuous Random Variables with CDFs

Analysis and Problem Solving

• Generate random samples from given distribution using inverse CDF method
• Derive statistical measures (expected values, variances) through integration techniques
• Analyze transformations of random variables, particularly for monotonic functions
• Determine stochastic dominance between different random variables or distributions
• Represent failure distribution in reliability analysis, with complement (1 - F(x)) as reliability function
• Utilize in hypothesis testing (Kolmogorov-Smirnov tests) to compare sample distributions to theoretical distributions
• Analyze relationships between multiple random variables using joint CDFs and marginal CDFs in multivariate settings

Applications in Various Fields

• Economics calculate income distributions and inequality measures (Gini coefficient)
• Finance model asset returns and price options (Black-Scholes model)
• Engineering analyze system reliability and component lifetimes (Weibull distribution)
• Physics study particle decay times and radioactive decay (exponential distribution)
• Environmental science model extreme weather events (generalized extreme value distribution)
• Quality control determine process capability and tolerance limits (normal distribution)
• Actuarial science assess insurance risks and set premiums (Pareto distribution)