Cumulative Distribution Functions (CDFs) are key tools for understanding random variables. They show the probability of a variable being less than or equal to a specific value, ranging from 0 to 1 and increasing monotonically.

CDFs bridge discrete and continuous random variables, allowing for probability calculations between points. They're essential for finding percentiles, generating random numbers, and analyzing complex systems with multiple variables.

Cumulative Distribution Function (CDF) Properties

Fundamental Characteristics of CDF

• Cumulative Distribution Function (CDF) represents the probability that a random variable takes on a value less than or equal to a given point
• Defines the probability distribution of a random variable X, denoted as F(x) = P(X ≤ x)
• Ranges from 0 to 1, with F(-∞) = 0 and F(∞) = 1
• Step Function characterizes CDFs for discrete random variables, jumps at each possible value of X
• Right-Continuous property ensures the function includes the endpoint of each interval

CDF Behavior and Applications

• Monotonically Increasing nature means F(x1) ≤ F(x2) for all x1 < x2
• Allows for calculation of probabilities between two points: P(a < X ≤ b) = F(b) - F(a)
• Inverse CDF, also known as the quantile function, finds the value of x for a given probability p
• Quantile Function proves useful in generating random numbers from a specific distribution
• Facilitates easy computation of median (50th percentile) and other percentiles of a distribution

Probability Functions and Random Variables

Comparing PDF and PMF

• Probability Density Function (PDF) applies to continuous random variables
• PDF represents the relative likelihood of a continuous random variable taking on a specific value
• Area under the PDF curve between two points gives the probability of the random variable falling within that range
• Probability Mass Function (PMF) pertains to discrete random variables
• PMF provides the probability of a discrete random variable taking on a specific value
• Sum of all probabilities in a PMF equals 1

Distinguishing Random Variable Types

• Discrete Random Variables take on countable, distinct values (dice rolls, number of customers)
• Continuous Random Variables can take any value within a given range (height, weight, time)
• Discrete variables use PMF, while continuous variables employ PDF
• CDF can be applied to both discrete and continuous random variables
• For discrete variables, CDF is a step function; for continuous variables, it's a smooth curve

Advanced CDF Concepts

Empirical and Multivariate CDFs

• Empirical CDF estimates the true CDF based on observed data points
• Constructs a step function that jumps by 1/n at each of the n data points
• Useful for non-parametric statistical inference and goodness-of-fit tests
• Joint CDF describes the probability distribution of two or more random variables simultaneously
• Denoted as F(x, y) = P(X ≤ x, Y ≤ y) for two random variables X and Y
• Allows for analyzing dependencies and correlations between multiple random variables

Deriving Univariate from Multivariate CDFs

• Marginal CDF focuses on the distribution of a single variable from a joint distribution
• Obtained by letting the other variables approach infinity in the joint CDF
• For two variables: FX(x) = lim(y→∞) F(x, y) and FY(y) = lim(x→∞) F(x, y)
• Enables studying individual variable behavior within a multivariate context
• Crucial for understanding relationships between variables in complex systems (financial markets, weather patterns)