Glossary

4.2 Cumulative distribution functions

3 min read•july 19, 2024

Cumulative Distribution Functions (CDFs) are key tools in probability theory. They show the likelihood of a random variable being less than or equal to a specific value, helping engineers calculate probabilities for various scenarios.

CDFs are derived from Probability Mass Functions (PMFs) for discrete variables. They have important properties like being non-decreasing and bounded between 0 and 1. CDFs are crucial for , , and assessments in engineering.

Cumulative Distribution Functions (CDFs)

Definition and role of CDFs

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  • denoted as F(x)F(x) represents the probability that a random variable XX takes a value less than or equal to xx
  • Mathematically expressed as F(x)=P(X≤x)F(x) = P(X \leq x), where PP denotes probability
  • Relates to for discrete random variables
    • PMF, denoted as p(x)p(x), gives the probability of a specific value of xx (rolling a 3 on a die)
    • CDF is the sum of PMF values for all values less than or equal to xx (probability of rolling a 3 or lower)
  • F(x)=∑k≤xp(k)F(x) = \sum_{k \leq x} p(k), where kk represents all possible values of the discrete random variable XX

Derivation of CDFs from PMFs

  • Derive the CDF from a given PMF by following these steps:
    1. Identify the possible values of the discrete random variable XX (0, 1, 2, 3 for a four-sided die)
    2. Calculate the PMF for each value of XX (p(0)=0.2p(0) = 0.2, p(1)=0.3p(1) = 0.3, p(2)=0.4p(2) = 0.4, p(3)=0.1p(3) = 0.1)
    3. Compute the CDF by summing the PMF values for all values less than or equal to xx
  • Example: Consider a discrete random variable XX with PMF: p(0)=0.2p(0) = 0.2, p(1)=0.3p(1) = 0.3, p(2)=0.4p(2) = 0.4, p(3)=0.1p(3) = 0.1
    • F(0)=P(X≤0)=p(0)=0.2F(0) = P(X \leq 0) = p(0) = 0.2
    • F(1)=P(X≤1)=p(0)+p(1)=0.2+0.3=0.5F(1) = P(X \leq 1) = p(0) + p(1) = 0.2 + 0.3 = 0.5
    • F(2)=P(X≤2)=p(0)+p(1)+p(2)=0.2+0.3+0.4=0.9F(2) = P(X \leq 2) = p(0) + p(1) + p(2) = 0.2 + 0.3 + 0.4 = 0.9
    • F(3)=P(X≤3)=p(0)+p(1)+p(2)+p(3)=0.2+0.3+0.4+0.1=1F(3) = P(X \leq 3) = p(0) + p(1) + p(2) + p(3) = 0.2 + 0.3 + 0.4 + 0.1 = 1

Probability calculations using CDFs

  • Calculate probabilities for discrete random variables using CDFs
    • P(a<X≤b)=F(b)−F(a)P(a < X \leq b) = F(b) - F(a), where aa and bb are values of the random variable XX (probability of rolling between a 2 and 4 on a six-sided die)
  • Engineering applications:
    • Reliability analysis calculates the probability of a system or component failing within a specific time frame (probability of a light bulb failing within 1000 hours)
    • Quality control determines the probability of a product meeting certain specifications (probability of a manufactured part being within tolerance)
    • Network performance assesses the probability of a network experiencing a certain level of congestion or delay (probability of a network having a delay greater than 100 ms)

Properties and significance of CDFs

  • Properties of CDFs:
    • Non-decreasing: F(x1)≤F(x2)F(x_1) \leq F(x_2) if x1<x2x_1 < x_2 (probability of rolling a 3 or lower is less than or equal to rolling a 4 or lower)
    • : lim⁡x→a+F(x)=F(a)\lim_{x \to a^+} F(x) = F(a) (CDF has no jumps at any point)
    • Bounds: 0≤F(x)≤10 \leq F(x) \leq 1 for all xx (probability is always between 0 and 1)
    • lim⁡x→−∞F(x)=0\lim_{x \to -\infty} F(x) = 0 and lim⁡x→∞F(x)=1\lim_{x \to \infty} F(x) = 1 (probability of any value occurring is 1)
  • Significance in probability calculations:
    • CDFs provide a comprehensive description of the probability distribution of a random variable
    • Enable the calculation of probabilities for any range of values (probability of a component lasting between 1000 and 2000 hours)
    • Allow for the comparison of different probability distributions (comparing failure rates of two different components)
    • Facilitate the computation of other statistical measures, such as expectation and variance (expected lifetime of a component)

Key Terms to Review (20)

Cumulative Distribution Function (CDF): The cumulative distribution function (CDF) of a random variable is a function that maps values to the probability that the variable takes on a value less than or equal to that number. It provides a complete description of the probability distribution and is essential in understanding properties such as expected value and variance for both discrete and continuous random variables. The CDF also helps in the analysis of probability density functions and plays a significant role in distributions like gamma and beta.
Distribution Comparison: Distribution comparison is the process of analyzing and contrasting two or more probability distributions to identify differences in their characteristics, such as their shapes, means, variances, and overall behavior. This concept plays a crucial role in understanding how different datasets relate to each other, helping to make informed decisions based on statistical data.
F(-∞) = 0: The expression f(-∞) = 0 refers to the property of a cumulative distribution function (CDF) that indicates the probability of a random variable being less than or equal to negative infinity is zero. This property is essential because it establishes a foundational aspect of how probabilities are assigned across the entire range of a random variable, ensuring that the total probability is properly normalized between the lower and upper bounds of the distribution.
F(∞) = 1: The expression f(∞) = 1 signifies that the cumulative distribution function (CDF) approaches a value of 1 as the variable approaches infinity. This indicates that as we consider all possible values of a random variable, the probability that the variable takes on any value within its support eventually accumulates to 1. This property reflects one of the key characteristics of probability distributions, ensuring that all probabilities are accounted for in the range of the random variable.
F(x) = ∑_{k ≤ x} p(k): This equation represents the cumulative distribution function (CDF) of a discrete random variable, where $f(x)$ is the probability that the random variable takes a value less than or equal to $x$. It is derived by summing the probabilities $p(k)$ of all outcomes $k$ that are less than or equal to $x$. The CDF provides a complete description of the distribution, as it encompasses all probabilities up to a certain point, allowing for an understanding of how the probabilities accumulate.
F(x) = p(x ≤ x): The expression f(x) = p(x ≤ x) represents the cumulative distribution function (CDF) of a random variable, showing the probability that the variable takes on a value less than or equal to a specific value x. This function is crucial because it helps in understanding the distribution of probabilities over the range of values for a given random variable, providing insights into its behavior. The CDF is defined for all real numbers and is non-decreasing, meaning as x increases, the probability does not decrease.
Limits at Infinity: Limits at infinity refer to the behavior of a function as the input values approach positive or negative infinity. This concept is crucial for understanding how functions behave in the long run, particularly in the context of probability distributions and cumulative distribution functions. It helps determine the tail behavior of distributions and is essential for evaluating probabilities of extreme events.
Mean: The mean, often referred to as the average, is a measure of central tendency that quantifies the expected value of a random variable. It represents the balancing point of a probability distribution, providing insight into the typical outcome one can expect from a set of data or a probability distribution. The concept of the mean is essential in understanding various statistical properties and distributions, as it lays the foundation for further analysis and interpretation.
Network performance: Network performance refers to the overall efficiency and effectiveness of a computer network in transmitting data and providing services. This encompasses various metrics such as bandwidth, latency, throughput, and error rates, which help assess how well the network meets the demands of its users. Understanding network performance is crucial for optimizing system reliability, minimizing delays, and ensuring high-quality user experiences.
Non-decreasing function: A non-decreasing function is a type of mathematical function where, as the input value increases, the output value does not decrease. In simpler terms, if you take any two points in the domain of the function, where the first point is less than or equal to the second point, the function's value at the first point is less than or equal to the value at the second point. This property is crucial in understanding cumulative distribution functions, which are inherently non-decreasing since they represent probabilities that accumulate as you move along the range of values.
Percentile: A percentile is a statistical measure that indicates the relative standing of a value within a data set, representing the percentage of observations that fall below that value. It helps to understand the distribution of data by dividing it into 100 equal parts, allowing comparisons across different data sets and highlighting the position of specific data points. This concept is crucial in interpreting cumulative distribution functions and understanding continuous random variables.
Probability Mass Function (pmf): A probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. The pmf provides a complete description of the distribution of a discrete random variable, showing how probabilities are allocated across different possible outcomes. It connects to important concepts like cumulative distribution functions, which aggregate probabilities, and maximum likelihood estimation, where it helps in estimating parameters based on observed data.
Probability of a product meeting specifications: The probability of a product meeting specifications refers to the likelihood that a manufactured product will satisfy predefined standards or requirements during its production process. This concept is crucial in quality control and reliability engineering, as it helps assess whether a product will function as intended and meet customer expectations. Understanding this probability is essential for making informed decisions about production processes, resource allocation, and risk management in engineering.
Probability of a system failing: The probability of a system failing refers to the likelihood that a given system will not perform its intended function within a specified period of time. This concept is crucial in evaluating the reliability and performance of various engineering systems, allowing engineers to make informed decisions regarding design, maintenance, and risk management. By understanding this probability, stakeholders can assess the risks involved and implement measures to mitigate potential failures.
Probability of Network Delay: The probability of network delay refers to the likelihood that a data packet transmitted over a network will experience a delay before reaching its destination. This concept is crucial as it directly affects the performance and reliability of network communications, making it essential to understand how delays can vary across different networks and conditions. It is often analyzed using statistical methods, where cumulative distribution functions help in modeling and predicting the behavior of such delays in various scenarios.
Quality Control: Quality control is a systematic process that ensures products or services meet specified requirements and standards. This process involves monitoring and evaluating various aspects of production and service delivery, using statistical methods to identify and correct deviations from desired quality levels. Effective quality control helps minimize defects, reduce costs, and increase customer satisfaction, making it essential in manufacturing and service industries.
Quantile: A quantile is a statistical value that divides a data set into equal-sized intervals, helping to describe the distribution of data points. Specifically, quantiles help in identifying specific percentages of data below or above a certain value, which can be useful for understanding the overall spread and central tendencies of a dataset. In relation to cumulative distribution functions, quantiles offer a way to express probabilities corresponding to specific values in the distribution.
Reliability Analysis: Reliability analysis is a statistical method used to assess the consistency and dependability of a system or component over time. It focuses on determining the probability that a system will perform its intended function without failure during a specified period under stated conditions. This concept is deeply interconnected with random variables and their distributions, as understanding the behavior of these variables is crucial for modeling the reliability of systems and processes.
Right-continuous: A function is right-continuous at a point if the limit of the function as it approaches that point from the right is equal to the function's value at that point. This concept is crucial in understanding how cumulative distribution functions (CDFs) behave, particularly in defining the properties of distributions and their continuity. Right-continuity ensures that there are no jumps or breaks in the function at any given point when approaching from the right, making it a fundamental aspect when dealing with probabilities and continuous random variables.
Stochastic dominance: Stochastic dominance is a concept used in decision theory and economics to compare different probability distributions, where one distribution is considered better than another based on expected utility. It provides a method to evaluate risky options by assessing which option will yield higher expected outcomes for all levels of utility. This concept relies heavily on cumulative distribution functions (CDFs), which illustrate the probability that a random variable takes a value less than or equal to a certain threshold, making it crucial for understanding risk and choice under uncertainty.
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