Glossary

3.2 Cumulative distribution functions

4 min readâ€ĸaugust 16, 2024

Cumulative distribution functions (CDFs) are essential tools in probability theory. They provide a complete description of a random variable's probability distribution, allowing us to calculate probabilities for various events and understand the overall behavior of the variable.

CDFs have unique properties that make them valuable for both discrete and continuous random variables. They're non-decreasing functions that range from 0 to 1, approaching these limits as x goes to negative and positive infinity, respectively. This makes CDFs incredibly useful for comparing different distributions and solving complex probability problems.

Cumulative Distribution Functions

Definition and Key Properties

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  • (CDF) F(x) for random variable X defined as F(x)=P(X≤x)F(x) = P(X ≤ x) for all real numbers x
  • where F(x1)≤F(x2)F(x₁) ≤ F(x₂) for all x1<x2x₁ < x₂
  • Range between 0 and 1 0≤F(x)≤10 ≤ F(x) ≤ 1 for all x
  • Approaches 0 as x approaches negative infinity lim⁥x→−∞F(x)=0\lim_{x \to -\infty} F(x) = 0
  • Approaches 1 as x approaches positive infinity lim⁥x→+∞F(x)=1\lim_{x \to +\infty} F(x) = 1
  • Continuous for continuous random variables, for discrete random variables
  • Right-continuous lim⁥h→0+F(x+h)=F(x)\lim_{h \to 0^+} F(x+h) = F(x) for all x
    • Ensures well-defined probabilities at discontinuities
    • Important for theoretical properties and proofs

Characteristics for Different Types of Random Variables

  • Discrete random variables
    • CDF takes form of step function
    • Jumps occur at possible values of X
    • Size of jump at x equals P(X = x)
    • Example: For a fair six-sided die, CDF jumps by 1/6 at each integer from 1 to 6
  • Continuous random variables
    • CDF smooth, continuous curve
    • No jumps or discontinuities
    • Slope of CDF at x gives probability density at that point
    • Example: For standard , CDF smoothly increases from 0 to 1, with steepest slope around x = 0

Calculating Probabilities with CDFs

Basic Probability Calculations

  • Probability X less than or equal to a P(X≤a)=F(a)P(X ≤ a) = F(a)
  • Probability X greater than a P(X>a)=1−F(a)P(X > a) = 1 - F(a)
  • Probability X between a and b (a < b) P(a<X≤b)=F(b)−F(a)P(a < X ≤ b) = F(b) - F(a)
  • Discrete random variables P(X=a)=F(a)−lim⁥x→a−F(x)P(X = a) = F(a) - \lim_{x \to a^-} F(x)
    • Example: For a geometric distribution with p = 0.3, P(X=2)=F(2)−F(1)P(X = 2) = F(2) - F(1)
  • Continuous random variables P(X=a)=0P(X = a) = 0 for any specific point a
    • Example: For a uniform distribution on [0,1], P(X=0.5)=0P(X = 0.5) = 0

Advanced Applications

  • Median found by solving F(x)=0.5F(x) = 0.5
  • calculated by finding inverse of CDF at desired probability
    • Example: 75th percentile is x where F(x)=0.75F(x) = 0.75
  • Quantile function Q(p) inverse of CDF, gives value x where F(x)=pF(x) = p
    • Used in generating random variables from uniform distributions
  • Expected value can be computed using CDF E[X]=âˆĢ0∞(1−F(x))dx−âˆĢ−∞0F(x)dxE[X] = \int_{0}^{\infty} (1 - F(x)) dx - \int_{-\infty}^{0} F(x) dx
    • Useful when PDF or PMF not directly available

PMFs vs CDFs

Relationship for Discrete Random Variables

  • CDF sum of PMF values up to and including x F(x)=∑k≤xP(X=k)F(x) = \sum_{k \leq x} P(X = k)
  • PMF derived from CDF P(X=x)=F(x)−F(x−1)P(X = x) = F(x) - F(x-1)
  • Example: For Poisson distribution with Îģ = 2, CDF at x = 3 sum of PMF values for k = 0, 1, 2, 3
  • Useful for finding probabilities of ranges of values
    • P(a<X≤b)=F(b)−F(a)P(a < X ≤ b) = F(b) - F(a) more efficient than summing individual PMF values

Relationship for Continuous Random Variables

  • (PDF) derivative of CDF f(x)=ddxF(x)f(x) = \frac{d}{dx} F(x)
  • CDF integral of PDF F(x)=âˆĢ−∞xf(t)dtF(x) = \int_{-\infty}^{x} f(t)dt
  • Example: For with rate Îģ, CDF F(x)=1−e−ÎģxF(x) = 1 - e^{-Îģx}, PDF f(x)=Îģe−Îģxf(x) = Îģe^{-Îģx}
  • Relationship allows conversion between PDF and CDF representations
    • Useful for solving problems involving both discrete and continuous random variables
  • Area under PDF curve between two points equals probability in that interval
    • Corresponds to difference in CDF values at those points

Interpreting CDFs Graphically

Visual Characteristics

  • Non-decreasing function starting at 0 for x approaching negative infinity, approaching 1 for x approaching positive infinity
  • Discrete random variables CDF step function with jumps at each possible value of X
    • Example: Binomial distribution with n = 5, p = 0.5 has jumps at integers 0 through 5
  • Continuous random variables CDF smooth, continuous curve
    • Example: Normal distribution CDF S-shaped curve, symmetric around mean
  • Steepness of CDF curve indicates concentration of probability in that region
    • Steeper sections correspond to higher probability density
  • Horizontal sections represent gaps in support of distribution (zero probability areas)
    • Example: Discrete uniform distribution on {1, 3, 5} has horizontal sections between jumps

Extracting Information from CDF Graphs

  • Y-intercept of vertical line at x-value represents probability P(X≤x)P(X ≤ x)
  • Difference in y-values between two points probability of X falling between those x-values
  • Shape of CDF provides insights into distribution's properties
    • Symmetric distributions have CDF symmetric around median
    • Skewed distributions have asymmetric CDFs
  • Inflection points in continuous CDFs indicate modes of distribution
  • Vertical distance between two CDFs at given x-value shows difference in probabilities
    • Useful for comparing different distributions or parameters

Key Terms to Review (16)

Cdf graph: A cumulative distribution function (CDF) graph is a visual representation that shows the probability that a random variable takes on a value less than or equal to a specific value. This graph provides insight into the distribution of probabilities and helps to understand how values accumulate over time, allowing for easy identification of probabilities associated with various intervals.
Continuous Cumulative Distribution Function: A continuous cumulative distribution function (CDF) is a mathematical function that describes the probability that a continuous random variable will take a value less than or equal to a certain value. This function is essential in probability theory as it provides a complete description of the distribution of the random variable, allowing for the calculation of probabilities over intervals and the understanding of the variable's behavior across its range.
Convolution: Convolution is a mathematical operation that combines two functions to produce a third function, representing how the shape of one is modified by the other. In the context of probability theory, convolution is particularly useful for finding the distribution of the sum of two independent random variables by integrating the product of their probability density functions. This operation links to cumulative distribution functions by illustrating how these functions can be derived from the convolution of individual distributions.
Cumulative Distribution Function: A cumulative distribution function (CDF) is a mathematical function that describes the probability that a random variable takes on a value less than or equal to a specified value. It provides a complete description of the probability distribution, whether for discrete or continuous random variables, and is fundamental in understanding how probabilities accumulate over intervals.
Discrete Cumulative Distribution Function: A discrete cumulative distribution function (CDF) is a mathematical function that gives the probability that a discrete random variable is less than or equal to a certain value. This function helps to summarize the probabilities of all possible outcomes in a discrete sample space, providing insight into how the probabilities accumulate as one moves through the potential values of the random variable. It essentially serves as a tool to visualize and analyze the distribution of probabilities across distinct outcomes.
Exponential Distribution: The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process. It is characterized by its memoryless property, meaning the probability of an event occurring in the future is independent of any past events, which connects it to processes where events occur continuously and independently over time.
Fundamental Theorem of Probability: The Fundamental Theorem of Probability establishes a foundational relationship between cumulative distribution functions (CDFs) and probability measures, indicating that the probability of an event can be derived from the CDF. It connects the way probabilities are assigned to outcomes with how they accumulate across different intervals, emphasizing the importance of understanding distribution functions in probability theory.
Inverse Transform Sampling: Inverse transform sampling is a method used to generate random samples from a probability distribution by utilizing the cumulative distribution function (CDF) of that distribution. This technique involves taking a uniformly distributed random variable and transforming it through the inverse of the CDF to produce samples that follow the desired distribution. The connection between the CDF and inverse transform sampling is crucial, as it allows for the conversion of uniform random variables into variables that follow more complex distributions.
Limits at infinity: Limits at infinity refer to the behavior of a function as the input values approach infinity or negative infinity. This concept helps to understand how functions behave in the long term, providing insights into horizontal asymptotes and the end behavior of functions in probability distributions, particularly in relation to cumulative distribution functions.
Non-decreasing function: A non-decreasing function is a type of function where the value does not decrease as the input increases, meaning if $x_1 \leq x_2$, then $f(x_1) \leq f(x_2)$. This property ensures that the function either stays constant or increases with increasing inputs. Non-decreasing functions are important because they help in defining cumulative distribution functions, which represent probabilities and must adhere to this characteristic to reflect a valid probability distribution.
Normal Distribution: Normal distribution is a probability distribution that is symmetric about the mean, representing the distribution of many types of data. Its shape is characterized by a bell curve, where most observations cluster around the central peak, and probabilities for values further away from the mean taper off equally in both directions. This concept is crucial because it helps in understanding how random variables behave and is fundamental to many statistical methods.
Percentiles: Percentiles are measures that indicate the relative standing of a value within a dataset, dividing the data into 100 equal parts. They help to understand how a particular score compares to others in the same dataset. For instance, if a score falls at the 70th percentile, it means that the score is higher than 70% of the values in the dataset. Percentiles are particularly useful in analyzing continuous distributions, probability density functions, and cumulative distribution functions to summarize data and make informed decisions based on statistical analysis.
Probability Density Function: A probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value. The PDF is integral in determining probabilities over intervals and is closely linked to cumulative distribution functions, expectation, variance, and various common distributions like uniform, normal, and exponential. It helps in understanding the behavior of continuous random variables by providing a framework for calculating probabilities and expectations.
Quantiles: Quantiles are values that divide a probability distribution into equal intervals, with each interval containing a specific proportion of the data. They help summarize the distribution of data points by indicating thresholds at which a certain percentage of the observations fall below. Understanding quantiles is crucial in analyzing various continuous distributions, such as uniform, normal, and exponential, as they provide insights into the behavior and characteristics of these distributions through their probability density and cumulative distribution functions.
Step Function: A step function is a piecewise constant function that takes only a finite number of values, with jumps at specific points. It is often used to represent cumulative distribution functions, which are non-decreasing and can exhibit sudden increases at discrete points corresponding to outcomes in a probability distribution.
Transformation: In probability theory, a transformation refers to the mathematical operation that modifies a random variable into another variable through a function. This process allows us to derive new distributions and analyze how the behavior of one variable influences another, making it crucial for understanding the relationships between random variables and their cumulative distribution functions.
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