Glossary

12.3 Probability Distributions

3 min read•august 12, 2024

Probability distributions are the backbone of discrete probability. They help us understand how random events occur and predict outcomes. From coin flips to website traffic, these models give us a way to make sense of uncertainty.

Bernoulli, binomial, geometric, and Poisson distributions tackle different scenarios in discrete probability. Meanwhile, uniform and normal distributions handle continuous cases. Understanding these helps us solve real-world problems and make informed decisions.

Discrete Probability Distributions

Bernoulli and Binomial Distributions

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  • models a single trial with two possible outcomes (success or failure)
    • Probability of success denoted as p, probability of failure as 1-p
    • Random variable X takes value 1 with probability p and 0 with probability 1-p
    • given by P(X=k)=pk(1−p)1−kP(X=k) = p^k(1-p)^{1-k} for k = 0 or 1
    • () of Bernoulli distribution equals p
    • of Bernoulli distribution equals p(1-p)
  • extends Bernoulli to n independent trials
    • Models number of successes in n trials with probability p of success on each trial
    • Probability mass function given by P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k}p^k(1-p)^{n-k} for k = 0, 1, ..., n
    • Mean of binomial distribution equals np
    • Variance of binomial distribution equals np(1-p)
  • Applications include coin flips (Bernoulli) and number of heads in multiple coin flips (binomial)

Geometric and Poisson Distributions

  • models number of trials until first success
    • Probability mass function given by P(X=k)=(1−p)k−1pP(X=k) = (1-p)^{k-1}p for k = 1, 2, 3, ...
    • Mean of geometric distribution equals 1/p
    • Variance of geometric distribution equals (1-p)/p^2
    • Used in scenarios like number of attempts until passing an exam
  • models number of events in a fixed interval
    • Parameter λ represents average number of events in the interval
    • Probability mass function given by P(X=k)=e−λλkk!P(X=k) = \frac{e^{-λ}λ^k}{k!} for k = 0, 1, 2, ...
    • Mean and variance of Poisson distribution both equal λ
    • Applications include modeling rare events (radioactive decay, website traffic spikes)

Continuous Probability Distributions

Uniform and Normal Distributions

  • represents constant probability over an interval [a,b]
    • given by f(x)=1b−af(x) = \frac{1}{b-a} for a ≤ x ≤ b
    • Mean of uniform distribution equals (a+b)/2
    • Variance of uniform distribution equals (b-a)^2/12
    • Used in scenarios with equal likelihood (rolling a fair die)
  • (Gaussian distribution) characterized by bell-shaped curve
    • Defined by parameters μ (mean) and σ ()
    • Probability density function given by f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{σ\sqrt{2π}}e^{-\frac{(x-μ)^2}{2σ^2}}
    • Standard normal distribution has μ = 0 and σ = 1
    • Widely used in natural and social sciences (height distribution, measurement errors)

Cumulative Distribution Function

  • (CDF) gives probability of a random variable being less than or equal to a value
    • For discrete distributions, CDF is sum of probabilities up to that value
    • For continuous distributions, CDF is integral of probability density function up to that value
  • Properties of CDF:
    • Monotonically increasing function
    • Limits: F(-∞) = 0 and F(∞) = 1
    • Right-continuous for discrete distributions
  • Relationship between probability density function (PDF) and CDF:
    • For continuous distributions, PDF is derivative of CDF
    • CDF is integral of PDF: F(x)=∫−∞xf(t)dtF(x) = \int_{-∞}^x f(t)dt
  • Applications include calculating probabilities for intervals and finding percentiles

Key Terms to Review (21)

Bernoulli Distribution: The Bernoulli distribution is a discrete probability distribution for a random variable which has exactly two possible outcomes, typically referred to as 'success' and 'failure'. It is a special case of the binomial distribution where a single trial is conducted, making it foundational in probability theory and statistics.
Binomial distribution: A binomial distribution is a probability distribution that summarizes the likelihood of a certain number of successes in a fixed number of independent trials, each with the same probability of success. This distribution is characterized by two outcomes, typically referred to as 'success' and 'failure,' and is described by parameters: the number of trials (n) and the probability of success (p). The connection to random variables and expectation comes from its ability to model discrete random events, while its role in probability distributions is to provide a framework for calculating the probabilities of various outcomes.
Central Limit Theorem: The Central Limit Theorem states that when independent random variables are added, their normalized sum tends toward a normal distribution, regardless of the original distribution of the variables, as long as the sample size is sufficiently large. This theorem is crucial because it allows for the use of normal distribution in making inferences about population parameters based on sample statistics.
Continuous random variable: A continuous random variable is a type of variable that can take on an infinite number of values within a given range. This concept is crucial in probability theory as it represents measurements that can vary smoothly and can include any value within an interval. Understanding continuous random variables allows for the exploration of distributions and expectations related to outcomes that are not discrete, thereby enabling the application of various statistical methods.
Cumulative Distribution Function: The 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 specific value. It provides a complete description of the probability distribution of a random variable, whether it's discrete or continuous, and plays a crucial role in understanding the behavior and properties of random variables. The CDF allows us to determine probabilities and can be derived from probability density functions or probability mass functions.
Discrete Random Variable: A discrete random variable is a numerical outcome of a random phenomenon that can take on a countable number of distinct values, such as integers. It is fundamentally important in understanding how probabilities are assigned to different outcomes and plays a crucial role in the calculation of expected values and probability distributions. By analyzing discrete random variables, we can derive important statistical measures, make predictions, and interpret data effectively.
Expected Value: Expected value is a fundamental concept in probability that represents the average outcome of a random variable when considering all possible values and their associated probabilities. It provides a way to quantify the long-term average if an experiment or process is repeated many times, allowing for informed decision-making under uncertainty. The expected value can be calculated for both discrete and continuous random variables, highlighting its importance in various probability distributions.
Geometric distribution: Geometric distribution is a probability distribution that models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials, where each trial has the same probability of success. It captures scenarios where you're interested in how long it takes for an event to occur, emphasizing the concept of waiting time until success. This distribution is closely tied to random variables and the expectation of outcomes related to probabilistic events.
Kurtosis: Kurtosis is a statistical measure that describes the distribution of data points in a dataset, particularly the tails and the peak of the distribution. It provides insight into the extremity of values and how they compare to a normal distribution, indicating whether the data has heavy tails (more outliers) or light tails (fewer outliers). Understanding kurtosis helps to assess the risk and behavior of data, especially in probability distributions.
Law of large numbers: The law of large numbers is a fundamental theorem in probability that states as the number of trials in an experiment increases, the sample mean will converge to the expected value. This principle is key when discussing probability distributions, as it emphasizes the relationship between probability and long-term outcomes. Essentially, it assures that with a sufficient number of observations, the average result will be very close to the expected value, reducing the impact of random fluctuations.
Mean: The mean is a statistical measure that represents the average value of a set of numbers, calculated by dividing the sum of all values by the total number of values. In the context of probability distributions, the mean serves as a measure of central tendency, providing insights into the expected outcome of a random variable. It plays a crucial role in understanding how data is distributed and is often referred to as the expected value in probability theory.
Normal distribution: Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. This concept is crucial in statistics because it describes how the values of a variable are distributed and is often represented as a bell-shaped curve. Many statistical tests assume that the data follows a normal distribution, making it foundational for understanding probability and data analysis.
Poisson distribution: The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. This distribution is particularly useful for modeling rare events, such as the number of accidents at a traffic intersection in a day or the number of emails received per hour. Understanding this distribution helps in assessing expectations and probabilities related to random variables.
Probability Density Function: A probability density function (PDF) is a statistical function that describes the likelihood of a continuous random variable taking on a particular value. It provides a way to characterize the distribution of continuous outcomes, allowing the calculation of probabilities for ranges of values rather than specific outcomes. The integral of the PDF over a certain interval gives the probability that the random variable falls within that interval, highlighting its importance in understanding continuous probability distributions.
Probability Mass Function: A probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. It assigns probabilities to each possible outcome of the discrete random variable, ensuring that the total probability across all outcomes sums to one. The PMF is crucial for understanding how probabilities are distributed across different values and forms the foundation for calculating expected values and assessing distributions.
Risk assessment: Risk assessment is the process of identifying, analyzing, and evaluating potential risks that could negatively impact an organization's objectives or projects. This process helps organizations understand the likelihood and impact of various risks, enabling them to make informed decisions about risk management strategies. By employing this systematic approach, organizations can prioritize risks and allocate resources more effectively to mitigate them.
Skewness: Skewness is a statistical measure that indicates the asymmetry of a probability distribution. It reflects the degree to which a distribution leans towards one side of the mean, with positive skewness indicating a longer tail on the right and negative skewness indicating a longer tail on the left. Understanding skewness is essential for interpreting data and making informed decisions based on its shape, as it can influence various statistical analyses and results.
Standard deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. It helps in understanding how spread out the numbers are in relation to the mean, providing insights into the consistency or variability of data. In probability theory, it plays a crucial role in determining how likely it is for a random variable to deviate from its expected value, while in probability distributions, it defines the shape and spread of the distribution itself.
Statistical inference: Statistical inference is the process of drawing conclusions about a population based on sample data. This method allows researchers to make predictions and decisions regarding the characteristics of a larger group without having to examine every single member of that group. By utilizing probability distributions, statistical inference enables one to estimate population parameters, test hypotheses, and assess the reliability of these conclusions.
Uniform Distribution: Uniform distribution is a type of probability distribution where all outcomes are equally likely to occur. This means that each value within a defined range has the same probability of being selected, leading to a flat probability density function. This concept is crucial when analyzing random variables and understanding how probabilities are distributed across different events.
Variance: Variance is a statistical measurement that indicates the degree to which data points in a set differ from the mean of that set. It provides insight into the spread or dispersion of data, revealing how much individual values vary from the average value. Understanding variance is crucial for analyzing data distributions, making predictions, and assessing the reliability of statistical conclusions.
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