4.6 Poisson Distribution (Optional)

The Poisson distribution is a discrete probability distribution that describes the number of events occurring in a fixed interval of time or space, given that these events happen with a known average rate and independently of the time since the last event. It is commonly used to model rare events that occur randomly and independently over time or space.

Discrete Random Variable: A random variable that can take on a countable number of distinct values, often integers, rather than a continuous range of values.

Probability Mass Function (PMF): The function that gives the probability that a discrete random variable is exactly equal to some value.

Central Limit Theorem (CLT): A fundamental result in probability and statistics that states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the underlying distribution of the individual variables.

The law of rare events, also known as the Poisson distribution, describes 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 rate and independently of the time since the last event. This law is particularly useful in modeling and analyzing phenomena where rare or infrequent events occur, such as the number of phone calls received by a customer service center or the number of radioactive decays detected by a Geiger counter.

Poisson Distribution: A probability distribution that describes the number of events occurring in a fixed interval of time or space, given that these events occur with a known constant rate and independently of the time since the last event.

Rare Event: An event that occurs infrequently, with a low probability of happening within a given time or space.

Constant Rate: The rate at which rare events occur, which remains the same over the interval of time or space being considered.

The probability mass function (PMF) is a function that gives the probability that a discrete random variable will take on a particular value. It is a fundamental concept in probability theory and statistics, particularly in the context of discrete probability distributions such as the Geometric Distribution and Poisson Distribution.

Discrete Random Variable: A random variable that can only take on a countable number of distinct values, as opposed to a continuous random variable which can take on any value within a range.

Probability Distribution: A function that gives the probabilities of the different possible outcomes of a random variable.

Cumulative Distribution Function (CDF): The function that gives the probability that a random variable is less than or equal to a given value.

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes: success or failure. It is a fundamental concept in probability theory and statistics, with applications across various fields.

Bernoulli Trial: A Bernoulli trial is a probabilistic experiment with only two possible outcomes, typically labeled as 'success' and 'failure,' where the probability of success remains constant across all trials.

Discrete Random Variable: A discrete random variable is a variable that can take on a countable number of distinct values, as opposed to a continuous random variable that can take on any value within a range.

Probability Mass Function (PMF): The probability mass function (PMF) is a function that gives the probability that a discrete random variable will take on a particular value.

The Probability Mass Function (PMF) is a fundamental concept in probability theory that describes the probability distribution of a discrete random variable. It provides the probability of each possible value that the random variable can take, allowing for a comprehensive understanding of the likelihood of different outcomes in a discrete probability distribution.

Discrete Random Variable: A random variable that can only take on a countable number of distinct values, as opposed to a continuous random variable that can take on any value within a range.

Probability Distribution: A function that assigns probabilities to the possible values of a random variable, describing the likelihood of each outcome.

Cumulative Distribution Function (CDF): The function that gives the probability that a discrete random variable is less than or equal to a given value, providing a complementary perspective to the PMF.

The Cumulative Distribution Function (CDF) is a fundamental concept in probability and statistics that describes the probability of a random variable taking a value less than or equal to a specified value. It provides a comprehensive understanding of the distribution of a random variable and is widely used in various statistical analyses, including the Poisson distribution, uniform distribution, and exponential distribution.

Probability Density Function (PDF): The Probability Density Function (PDF) is a function that describes the relative likelihood of a random variable taking on a given value.

Quantile Function: The Quantile Function is the inverse of the Cumulative Distribution Function, which allows you to find the value of a random variable that corresponds to a given probability.

Random Variable: A random variable is a variable whose value is subject to variations due to chance or probability.

The mean, also known as the arithmetic mean or average, is a measure of central tendency that represents the central or typical value in a dataset. It is calculated by summing all the values in the dataset and dividing by the total number of values. The mean is a widely used statistic that provides information about the location or central tendency of a distribution.

Median: The median is the middle value in a dataset when the values are arranged in order from smallest to largest. It represents the central value that divides the dataset into two equal halves.

Mode: The mode is the value that appears most frequently in a dataset. It represents the most common or typical value in the distribution.

Standard Deviation: The standard deviation is a measure of the spread or dispersion of a dataset, indicating how much the values vary around the mean.

Variance is a statistical measure that quantifies the amount of variation or dispersion in a dataset. It represents the average squared deviation from the mean, providing a way to understand the spread or distribution of data points around the central tendency.

Standard Deviation: The square root of the variance, which provides a measure of the average distance of data points from the mean in the original units of the data.

Covariance: A measure of the joint variation between two random variables, indicating the degree to which they vary together.

Correlation: A standardized measure of the linear relationship between two random variables, ranging from -1 to 1, with 0 indicating no linear relationship.

A discrete probability distribution is a probability distribution that describes the probability of a discrete random variable taking on a specific value. It is a mathematical function that assigns probabilities to each possible outcome of a discrete random variable.

Probability Mass Function (PMF): The probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value.

Discrete Random Variable: A discrete random variable is a random variable that can take on only a countable number of distinct values, often integers.

Bernoulli Distribution: The Bernoulli distribution is a discrete probability distribution that takes value 1 with probability $p$ and value 0 with probability $1-p$, where $p$ is the success probability.

The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process, where events occur at a constant average rate. It is commonly used to describe the lifetimes of electronic components, the waiting times between customer arrivals, and other situations involving the occurrence of random events.

Poisson Process: A Poisson process is a model for a series of discrete events occurring continuously and independently at a constant average rate.

Continuous Probability Function: A continuous probability function is a function that describes the probability distribution of a continuous random variable.

Continuous Distribution: A continuous distribution is a probability distribution for a continuous random variable, where the variable can take on any value within a specified range.

The memoryless property, also known as the Markov property, is a fundamental characteristic of certain probability distributions and stochastic processes. It states that the future state of a system depends only on its current state and not on its past states or history.

Markov Chain: A Markov chain is a stochastic process that satisfies the memoryless property, where the probability of transitioning to a future state depends only on the current state and not the past states.

Exponential Distribution: The exponential distribution is a continuous probability distribution that exhibits the memoryless property, meaning the probability of an event occurring in the future is independent of the time since the last event.

Poisson Process: A Poisson process is a stochastic process that models the occurrence of events over time and has the memoryless property, where the probability of an event occurring in the future is independent of the time since the last event.