Types of Probability Distribution in 2025

Probability distributions are mathematical functions that describe the likelihood of different outcomes in a random experiment. They can be categorized into discrete and continuous distributions. Discrete distributions deal with random variables that take on a finite or countable number of values. Common examples include the binomial distribution, which models the number of successes in a fixed number of trials, and the Poisson distribution, which describes the number of events occurring in a fixed interval of time or space. 

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On the other hand, continuous distributions apply to random variables that can take any value within a given range. The normal distribution, often represented as a bell curve, is a key example used to model data symmetrically distributed around a mean, such as human heights. 

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Other continuous distributions, like the exponential distribution, model the time between independent events occurring at a constant rate (e.g., the time between arrivals at a service center). In contrast, the uniform distribution assumes all outcomes within a range have the same probability. Each of these distributions has specific characteristics and applications, helping statisticians model real-world phenomena accurately.

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What Is Probability?

Probability is a branch of mathematics that deals with the likelihood or chance of an event occurring. It quantifies uncertainty, helping us predict the chances of different outcomes in random experiments or processes. The probability of an event is always a number between 0 and 1, where 0 means the event will not occur, and 1 means the event will certainly occur. Probabilities can be expressed as fractions, decimals, or percentages.

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For example, when flipping a fair coin, the probability of landing heads is 0.5, as there are two possible outcomes (heads or tails), and each is equally likely. Similarly, the probability of drawing a red card from a standard deck of cards is 26/52, or 0.5, since there are 26 red cards in a deck.

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Probability is used in a wide range of fields, including statistics, finance, science, engineering, and artificial intelligence, to model uncertainty and make informed decisions. It is often represented by probability distributions, which describe the likelihood of different outcomes in a random experiment. The study of probability helps us understand and quantify randomness, making it an essential tool for analyzing and predicting real-world events.

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What Are Probability Distributions?

Probability distributions are mathematical functions that describe the likelihood of different outcomes for a random variable. They provide a way to model and quantify uncertainty, showing how the values of a random variable are spread across possible outcomes. Probability distributions can be classified into two main types: discrete and continuous.

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• Discrete probability distributions deal with random variables that take on a countable number of distinct values. For example, the binomial distribution models the number of successes in a fixed number of trials, and the Poisson distribution describes the number of events occurring in a fixed interval of time or space. In discrete distributions, the sum of the probabilities of all possible outcomes is always equal to 1.

• Continuous probability distributions apply to random variables that can take any value within a given range. The normal distribution, often referred to as the bell curve, is one of the most well-known continuous distributions, describing data that tends to cluster around a mean value. Other examples include the exponential distribution, which models the time between events in a process, and the uniform distribution, where every outcome in a given range has an equal probability.

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Probability distributions are fundamental in statistics, as they help predict and analyze the likelihood of events, inform decision-making, and model real-world phenomena across various fields.

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Types of Probability Distribution

Probability distributions are fundamental concepts in statistics and probability theory that describe how the values of a random variable are distributed. They provide a way to model uncertainty and predict the likelihood of different outcomes in various situations.

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There are two main types of probability distributions: discrete distributions, which deal with countable outcomes, and continuous distributions, which handle uncountable, infinite outcomes.

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Each distribution has unique characteristics and is suited for different types of data and real-world scenarios, such as modeling the number of successes in trials, the time between events, or the distribution of data points around a mean. Understanding these distributions is essential for analyzing data and making informed decisions in fields like science, finance, and engineering.

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1. Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure) with a constant probability of success. It is commonly used in scenarios like coin tosses, quality control testing, or survey responses.

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The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p) in each trial. It is useful for modeling discrete data with two outcomes.

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2. Poisson Distribution

The Poisson distribution describes the number of events that occur in a fixed interval of time or space, assuming these events happen at a constant average rate and independently of one another.

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It is often used to model rare events, like the number of phone calls received at a call center, accidents occurring at a particular intersection, or emails arriving in an inbox. The distribution is defined by the rate (λ), which is the average number of events in the interval.

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3. Normal Distribution

The normal distribution, also known as the Gaussian distribution, is one of the most important continuous probability distributions. It describes data that is symmetrically distributed around a mean value, with the data points tapering off towards the extremes.

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This distribution is characterized by two parameters: the mean (µ) and the standard deviation (σ). It’s widely used in natural and social sciences, as many real-world phenomena, such as heights, test scores, or errors in measurements, tend to follow a normal distribution.

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4. Exponential Distribution

The exponential distribution models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is commonly used to model waiting times, such as the time between the arrival of customers at a service desk or the lifespan of a light bulb.

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The exponential distribution is characterized by the rate parameter (λ), which represents the inverse of the average time between events. It’s often used in survival analysis and reliability testing.

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5. Uniform Distribution

In a uniform distribution, all outcomes are equally likely within a defined range. It’s a type of continuous distribution where the probability of the variable taking any value in the interval is constant. For example, the outcome of rolling a fair die or selecting a random number between 0 and 1 follows a uniform distribution.

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The uniform distribution is characterized by two parameters: the minimum (a) and maximum (b) values. It is used in scenarios where there is no bias toward any specific outcome.

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6. Gamma Distribution

The gamma distribution is a continuous probability distribution often used to model waiting times or the time until a certain number of events occur in a Poisson process. It is characterized by two parameters: the shape parameter (k) and the rate parameter (λ).

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The gamma distribution generalizes the exponential distribution, which is a special case when k = 1. It is commonly used in queuing theory, insurance risk modeling, and in processes where events happen at varying rates.

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7. Log-Normal Distribution

The log-normal distribution describes a random variable whose logarithm is normally distributed. This distribution is used to model variables that are positively skewed, such as income, stock prices, or the size of particles in a material.

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If a quantity follows a log-normal distribution, its values are skewed to the right, meaning there’s a higher probability of smaller values, with a long tail of larger values. The mean and variance of the underlying normal distribution characterize it.

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8. Bernoulli Distribution

The Bernoulli distribution is a discrete probability distribution that models a random experiment with exactly two outcomes: success (1) or failure (0). It is the simplest discrete distribution and is characterized by a single parameter, the probability of success (p).

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Bernoulli distributions are foundational in probability theory, serving as the building block for other distributions like the binomial distribution. It's commonly used in scenarios like flipping a coin or passing a test.

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9. Beta Distribution

The beta distribution is a continuous probability distribution defined on the interval [0, 1], and it is used to model random variables that represent probabilities or proportions. It is characterized by two shape parameters, α (alpha) and β (beta), which determine the shape of the distribution.

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The beta distribution is often used in Bayesian statistics, modeling the distribution of random variables like success rates in experimental trials or proportions in survey data.

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10. Chi-Square Distribution

The chi-square distribution is a special case of the gamma distribution. It is used primarily in hypothesis testing, particularly in tests for goodness of fit and independence in contingency tables.

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It is defined by a single parameter, the degrees of freedom (df), which determines the shape of the distribution. The chi-square distribution is skewed to the right, and as the degrees of freedom increase, it approaches a normal distribution. It is used in various statistical tests, such as the chi-square test for independence.

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11. Student's t-Distribution

The Student's t-distribution is a continuous probability distribution used for hypothesis testing when the sample size is small, and the population variance is unknown. It is similar to the normal distribution but with heavier tails, which allows for more variability in the data.

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The t-distribution is characterized by the degrees of freedom (df), and it is particularly useful in estimating the mean of a normally distributed population when sample sizes are small (typically less than 30).

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12. Multinomial Distribution

The multinomial distribution is a generalization of the binomial distribution used when there are more than two possible outcomes. It models the probability of obtaining a set of outcomes in a fixed number of trials, where each trial can result in one of several outcomes, and the trials are independent.

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For example, it can be used to model the distribution of votes among different candidates in an election or the distribution of categories in a survey with multiple response options.

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Probability Distribution of Random Variables

The probability distribution of random variables describes how the probabilities of different outcomes are assigned to a random variable. A random variable is a variable whose value is determined by the outcome of a random process or experiment. There are two types of random variables, each with its corresponding probability distribution:

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1. Discrete Random Variables: These take on a countable number of distinct values. The probability mass function (PMF) gives the probability that a discrete random variable takes a specific value. For example, when rolling a die, the random variable (the die's outcome) takes values from 1 to 6, with equal probability (1/6 for each face). Common discrete distributions include:

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• Binomial distribution: Models the number of successes in a fixed number of independent trials.

• Poisson distribution: Models the number of events occurring in a fixed interval of time or space.

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2. Continuous Random Variables: These can take any value within a given range or interval, and the probability density function (PDF) describes the likelihood of a continuous random variable taking a specific value. The probability of any exact value is zero; instead, probabilities are computed over intervals. Common continuous distributions include:

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• Normal distribution: Describes data that is symmetrically distributed around the mean.

• Exponential distribution: Models the time between events in a Poisson process.

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In both cases, the sum (for discrete) or the integral (for continuous) of the probabilities over all possible values must equal 1, ensuring that one of the outcomes must occur. Probability distributions are essential in statistical modeling and hypothesis testing, helping to predict and analyze random phenomena.

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Probability Distribution Formulas

Probability distribution formulas are mathematical expressions that help calculate the probabilities of various outcomes for different types of random variables. Here are the key formulas for common probability distributions:

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1. Binomial Distribution

The binomial distribution describes the probability of getting exactly k successes in n independent trials where the probability of success on each trial is p.

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Formula:

P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k

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Where:

• P(X=k)P(X = k)P(X=k) is the probability of exactly k successes.

• (nk)\binom{n}{k}(kn​) is the binomial coefficient, calculated as n!k!(n−k)!\frac{n!}{k!(n-k)!}k!(n−k)!n!​.

• ppp is the probability of success.

• (1−p)(1-p)(1−p) is the probability of failure.

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2. Poisson Distribution

The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, where the events happen at a constant rate.

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Formula:

P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}P(X=k)=k!λke−λ​

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Where:

• P(X=k)P(X = k)P(X=k) is the probability of exactly k events occurring.

• λ\lambdaλ is the average number of events in the given interval.

• eee is Euler’s number (approximately 2.71828).

• k!k!k! is the factorial of kkk.

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3. Normal Distribution

The normal distribution is used for continuous data that is symmetrically distributed around a mean. The probability density function (PDF) is given by:

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Formula:

f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}f(x)=σ2π​1​e−2σ2(x−μ)2​

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Where:

• f(x)f(x)f(x) is the probability density function.

• μ\muμ is the mean of the distribution.

• σ\sigmaσ is the standard deviation.

• xxx is the variable for which you are calculating the probability density.

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4. Exponential Distribution

The exponential distribution is used to model the time between events in a Poisson process (events happening continuously and independently at a constant rate).

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Formula:

f(x)=λe−λxforx≥0f(x) = \lambda e^{-\lambda x} \quad \text{for} \quad x \geq 0f(x)=λe−λxforx≥0

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Where:

• f(x)f(x)f(x) is the probability density function.

• λ\lambdaλ is the rate parameter (the inverse of the mean time between events).

• xxx is the time between events.

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5. Uniform Distribution

The uniform distribution is used when all outcomes in a given range are equally likely. For a continuous uniform distribution:

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Formula:

f(x)=1b−afora≤x≤bf(x) = \frac{1}{b - a} \quad \text{for} \quad a \leq x \leq bf(x)=b−a1​fora≤x≤b

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Where:

• f(x)f(x)f(x) is the probability density function.

• aaa and bbb are the lower and upper bounds of the interval, respectively.

• xxx is the variable for which you are calculating the probability.

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For a discrete uniform distribution, the probability of any specific outcome is 1n\frac{1}{n}n1​, where nnn is the number of possible outcomes.

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6. Gamma Distribution

The gamma distribution is a two-parameter family of continuous probability distributions, often used for modeling waiting times in a Poisson process.

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Formula:

f(x;α,β)=xα−1e−x/βΓ(α)βαforx≥0f(x; \alpha, \beta) = \frac{x^{\alpha - 1} e^{-x / \beta}}{\Gamma(\alpha) \beta^\alpha} \quad \text{for} \quad x \geq 0f(x;α,β)=Γ(α)βαxα−1e−x/β​forx≥0

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Where:

• f(x;α,β)f(x; \alpha, \beta)f(x;α,β) is the probability density function.

• α\alphaα is the shape parameter.

• β\betaβ is the scale parameter.

• Γ(α)\Gamma(\alpha)Γ(α) is the Gamma function.

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7. Bernoulli Distribution

The Bernoulli distribution models a single trial with two possible outcomes (success or failure). It is a special case of the binomial distribution when n=1n = 1n=1.

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Formula:

P(X=x)=px(1−p)1−xP(X = x) = p^x (1-p)^{1-x}P(X=x)=px(1−p)1−x

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Where:

• P(X=x)P(X = x)P(X=x) is the probability of the random variable XXX taking value xxx (where x∈{0,1}x \in \{0, 1\}x∈{0,1}).

• ppp is the probability of success (when x=1x = 1x=1).

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8. Beta Distribution

The beta distribution is a continuous distribution defined on the interval [0, 1], commonly used to model probabilities or proportions.

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Formula:

f(x;α,β)=xα−1(1−x)β−1B(α,β)for0≤x≤1f(x; \alpha, \beta) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)} \quad \text{for} \quad 0 \leq x \leq 1f(x;α,β)=B(α,β)xα−1(1−x)β−1​for0≤x≤1

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Where:

• f(x;α,β)f(x; \alpha, \beta)f(x;α,β) is the probability density function.

• α\alphaα and β\betaβ are shape parameters.

• B(α,β)B(\alpha, \beta)B(α,β) is the Beta function.

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9. Chi-Square Distribution

The chi-square distribution is used primarily in hypothesis testing, especially in goodness-of-fit tests.

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Formula:

f(x;k)=x(k/2)−1e−x/22k/2Γ(k/2)forx≥0f(x; k) = \frac{x^{(k/2) - 1} e^{-x/2}}{2^{k/2} \Gamma(k/2)} \quad \text{for} \quad x \geq 0f(x;k)=2k/2Γ(k/2)x(k/2)−1e−x/2​forx≥0

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Where:

• f(x;k)f(x; k)f(x;k) is the probability density function.

• kkk is the degree of freedom.

• Γ(k/2)\Gamma(k/2)Γ(k/2) is the Gamma function.

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Cumulative Probability Distribution

A cumulative probability distribution (also known as a cumulative distribution function or CDF) describes the probability that a random variable takes a value less than or equal to a given point. In other words, it accumulates the probabilities of all outcomes up to a certain value, providing a way to understand how the probability "builds up" over the range of possible values for a random variable.

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For a discrete random variable, the cumulative distribution function (CDF) is calculated by summing the probabilities of all values less than or equal to a specific value.

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The formula for CDF of Discrete Random Variable:

F(x)=P(X≤x)=∑k=−∞xP(X=k)F(x) = P(X \leq x) = \sum_{k=-\infty}^{x} P(X = k)F(x)=P(X≤x)=k=−∞∑x​P(X=k)

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Where:

• F(x)F(x)F(x) is the cumulative probability up to xxx.

• P(X=k)P(X = k)P(X=k) is the probability that XXX takes the value kkk.

• The sum is over all values of kkk that are less than or equal to xxx.

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For a continuous random variable, the CDF is the integral of the probability density function (PDF) from the lowest value to the value of interest. The CDF for continuous variables gives the probability that the variable takes a value less than or equal to a specific point.

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The formula for CDF of Continuous Random Variable:

F(x)=P(X≤x)=∫−∞xf(t) dtF(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \, dtF(x)=P(X≤x)=∫−∞x​f(t)dt

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Where:

• F(x)F(x)F(x) is the cumulative probability up to xxx.

• f(t)f(t)f(t) is the probability density function (PDF).

• The integral sums the probability density from −∞-\infty−∞ to xxx.

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Key Properties of CDF:

1. Non-decreasing: The CDF never decreases as the probability accumulates.

2. Range: The value of a CDF always lies between 0 and 1, i.e., 0≤F(x)≤10 \leq F(x) \leq 10≤F(x)≤1.

3. Limits:

• As x→−∞x \to -\inftyx→−∞, F(x)→0F(x) \to 0F(x)→0.

• As x→∞x \to \inftyx→∞, F(x)→1F(x) \to 1F(x)→1.

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4. Right-continuous: The CDF is continuous from the right, meaning there are no jumps when you approach a particular value from the right-hand side.

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Discrete Probability Distribution

A discrete probability distribution is a statistical distribution that describes the probability of outcomes for a discrete random variable. A discrete random variable can take on a countable number of distinct values, such as the result of a dice roll, the number of heads in coin flips, or the number of customers arriving at a store. Discrete probability distributions are used to model situations where the set of possible outcomes is finite or countably infinite.

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Key Features of Discrete Probability Distributions:

• Countable Outcomes: The variable can only take on specific values (e.g., 0, 1, 2, 3, ...).

• Probability Mass Function (PMF): A discrete probability distribution is often represented by a probability mass function (PMF), which gives the probability of each possible outcome. The sum of the probabilities of all possible outcomes must equal 1.

• Non-negative Probabilities: Each probability must be greater than or equal to 0.

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General Formula for Discrete Probability Distribution:

For a discrete random variable XXX, the probability mass function P(X=xi)P(X = x_i)P(X=xi​) gives the probability of each value xix_ixi​:

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P(X=xi)≥0for allxiand∑iP(X=xi)=1P(X = x_i) \geq 0 \quad \text{for all} \quad x_i \quad \text{and} \quad \sum_{i} P(X = x_i) = 1P(X=xi​)≥0for allxi​andi∑​P(X=xi​)=1

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Where:

• P(X=xi)P(X = x_i)P(X=xi​) is the probability that XXX takes the value xix_ixi​.

• The sum of the probabilities for all possible outcomes xix_ixi​ must equal 1.

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Common Discrete Probability Distributions:

1. Binomial Distribution: The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

• Formula: P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k

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Where:

• nnn is the number of trials,

• ppp is the probability of success on a single trial,

• kkk is the number of successes.

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2. Poisson Distribution: The Poisson distribution models the number of events occurring in a fixed interval of time or space when the events occur independently and at a constant rate.

• Formula: P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}P(X=k)=k!λke−λ

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​Where:

• λ\lambdaλ is the average rate of occurrence (mean number of events),

• kkk is the number of events, and

• eee is Euler's number (approximately 2.71828).

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3. Geometric Distribution: The geometric distribution models the number of trials required to achieve the first success in a series of independent trials, each with the same probability of success.

• Formula: P(X=k)=(1−p)k−1pP(X = k) = (1 - p)^{k-1} pP(X=k)=(1−p)k−1p

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Where:

• ppp is the probability of success in each trial,

• kkk is the number of trials until the first success.

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4. Negative Binomial Distribution: The negative binomial distribution generalizes the geometric distribution to model the number of trials required to achieve r successes.

• Formula: P(X=k)=(k−1r−1)pr(1−p)k−rP(X = k) = \binom{k-1}{r-1} p^r (1 - p)^{k-r}P(X=k)=(r−1k−1​)pr(1−p)k−r

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Where:

• rrr is the number of successes,

• kkk is the number of trials required to achieve rrr successes,

• ppp is the probability of success in each trial.

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5. Uniform Distribution (Discrete): A discrete uniform distribution occurs when each outcome in a finite set of outcomes has the same probability of occurring. It is often used for random experiments like rolling a fair die or drawing a card from a well-shuffled deck.

• Formula: P(X=xi)=1nP(X = x_i) = \frac{1}{n}P(X=xi​)=n1

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​Where:

• nnn is the number of possible outcomes,

• Each outcome xix_ixi​ has an equal probability.

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What is Negative Binomial Distribution?

The Negative Binomial Distribution is a discrete probability distribution that models the number of trials required to achieve a specified number of successes in a sequence of independent and identically distributed Bernoulli trials (trials with two possible outcomes: success or failure).

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It generalizes the geometric distribution, which models the number of trials required to achieve the first success. It extends it to cases where you are interested in the number of trials needed to achieve r success.

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Key Features of the Negative Binomial Distribution:

• Trials: It counts the number of trials until a fixed number of successes occur.

• Success Probability: The probability of success on each trial is constant (denoted as ppp).

• Failures: The distribution also counts the number of failures (denoted as XXX) before the rrr-th success.

• Independent Trials: Each trial is independent, meaning the outcome of one trial does not affect the others.

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Formula:

The probability mass function (PMF) for the negative binomial distribution is given by:

P(X=k)=(k−1r−1)pr(1−p)k−rP(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}P(X=k)=(r−1k−1​)pr(1−p)k−r

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Where:

• XXX is the total number of trials required to achieve r success.

• kkk is the total number of trials (which must be at least rrr).

• rrr is the number of successes.

• ppp is the probability of success on each trial.

• (k−1r−1)\binom{k-1}{r-1}(r−1k−1​) is the binomial coefficient, representing the number of ways to arrange r−1r-1r−1 successes in k−1k-1k−1 trials.

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What is Poisson Probability Distribution?

The Poisson probability distribution is a discrete probability distribution that models the number of events occurring within a fixed interval of time or space under certain conditions. These events must occur independently and at a constant average rate.

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The Poisson distribution is particularly useful when the events are rare and we want to predict the likelihood of a given number of events happening in a specific interval.

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Key Features of the Poisson Distribution:

• Independence: The occurrence of one event does not affect the occurrence of another.

• Constant Rate: The average rate (denoted as λ\lambdaλ, called the rate parameter) at which events occur is constant over time or space.

• Discrete: The distribution is used to model discrete counts of events (e.g., number of accidents, number of emails received, number of customer arrivals).

• Rare Events: The Poisson distribution is often used for modeling rare events that happen independently in a fixed period, like earthquakes, machine breakdowns, or calls to a customer service center.

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The formula for the Poisson Distribution:

The Poisson probability mass function (PMF) gives the probability of observing k events in a fixed interval, given the average rate of occurrence λ\lambdaλ (the expected number of events).

P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}P(X=k)=k!λke−λ​

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Where:

• P(X=k)P(X = k)P(X=k) is the probability of observing exactly kkk events.

• λ\lambdaλ is the average rate (mean number of events) that occur in a fixed interval.

• kkk is the actual number of events that occur (it must be a non-negative integer: k=0,1,2,…k = 0, 1, 2, \dotsk=0,1,2,…).

• eee is Euler's number (approximately 2.71828).

• k!k!k! is the factorial of kkk, which is the product of all positive integers less than or equal to kkk.

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Probability Distribution Function

A Probability Distribution Function (PDF) describes how the probabilities of a random variable are distributed across its possible values. It gives the likelihood of each possible outcome for a random variable, helping to understand how the random variable behaves.

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There are two main types of probability distribution functions:

• Probability Mass Function (PMF) – Used for discrete random variables.

• Probability Density Function (PDF) – Used for continuous random variables.

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1. Probability Mass Function (PMF):

A PMF applies to discrete random variables. A discrete random variable takes on countable values, such as the number of heads in coin flips or the number of cars passing through a toll booth.

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The PMF assigns probabilities to each outcome, ensuring that:

• The probability of each outcome is non-negative: P(X=xi)≥0P(X = x_i) \geq 0P(X=xi​)≥0.

• The sum of the probabilities of all possible outcomes equals 1: ∑iP(X=xi)=1\sum_{i} P(X = x_i) = 1i∑​P(X=xi​)=1

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Example (Discrete Random Variable):

For a fair six-sided die, the PMF assigns a probability of 16\frac{1}{6}61​ to each possible outcome (1, 2, 3, 4, 5, or 6):

P(X=x)=16,x∈{1,2,3,4,5,6}P(X = x) = \frac{1}{6}, \quad x \in \{1, 2, 3, 4, 5, 6\}P(X=x)=61​,x∈{1,2,3,4,5,6}

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2. Probability Density Function (PDF):

A PDF applies to continuous random variables, which can take on an infinite number of possible values within a given range. The PDF does not give the probability of a single value but rather the probability density. The probability that the random variable falls within a particular range is obtained by integrating the PDF over that range.

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The key properties of a PDF are:

• The PDF is always non-negative: f(x)≥0f(x) \geq 0f(x)≥0.

• The total area under the curve of the PDF equals 1 (representing the total probability): ∫−∞∞f(x) dx=1\int_{-\infty}^{\infty} f(x) \, dx = 1∫−∞∞​f(x)dx=1

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To find the probability that a continuous random variable XXX lies within a specific range [a,b][a, b][a,b], we integrate the PDF over that range:

P(a≤X≤b)=∫abf(x) dxP(a \leq X \leq b) = \int_{a}^{b} f(x) \, dxP(a≤X≤b)=∫ab​f(x)dx

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Example (Continuous Random Variable):

For a normal distribution, the probability density function is given by:

f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}f(x)=σ2π​1​e−2σ2(x−μ)2​

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Where:

• μ\muμ is the mean,

• σ\sigmaσ is the standard deviation.

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This function gives the probability density at any given point xxx. To find the probability that XXX falls within a range, we integrate this function over that range.

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What is the Prior Probability?

Prior Probability (often referred to simply as the "prior") is a concept in Bayesian probability theory that represents the probability of an event or hypothesis before new evidence or data is considered. In other words, it is the initial estimate of the likelihood of a particular outcome, based on prior knowledge, experience, or assumptions before any observations or experiments are conducted.

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The prior probability is a foundational element in Bayesian inference, where the goal is to update beliefs about the probability of an event as new evidence becomes available.

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Key Features of Prior Probability:

• Represents Initial Belief: The prior probability encapsulates what is known or assumed about a situation before collecting any new data. This could be based on historical data, expert opinion, or logical reasoning.

• Subjective: The prior can be subjective because it often reflects personal beliefs or knowledge about an event. In some cases, the prior is based on a well-established theoretical framework, while in others, it might rely on less objective information.

• Foundation for Bayesian Updating: Prior probability serves as the starting point for Bayesian updating, which is the process of adjusting the probability estimate as new evidence is observed. Once new data is collected, the prior is updated to form the posterior probability.

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What is Posterior Probability?

Posterior Probability is the probability of a hypothesis or event being true after considering new evidence or data. It is a key concept in Bayesian probability theory and is the result of updating our initial beliefs (prior probability) with observed data using Bayes' Theorem.

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Key Features of Posterior Probability:

1. Updated Belief: The posterior probability represents the updated belief or estimate of the likelihood of a hypothesis after new evidence or information is incorporated.

2. Combination of Prior and Likelihood: It combines two pieces of information:

• Prior Probability: The initial belief about the hypothesis before any data was observed.

• Likelihood: The probability of the observed data given the hypothesis.

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3. Dynamic: Posterior probability is not static. It evolves as new data becomes available. It becomes the new "prior" for future updates, allowing for continuous refinement of belief.

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Mathematical Formula (Bayes' Theorem):

In Bayesian inference, the posterior probability P(H∣D)P(H | D)P(H∣D) is calculated using Bayes' Theorem:

P(H∣D)=P(D∣H)⋅P(H)P(D)P(H | D) = \frac{P(D | H) \cdot P(H)}{P(D)}P(H∣D)=P(D)P(D∣H)⋅P(H)​

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Where:

• P(H∣D)P(H | D)P(H∣D) is the posterior probability: The probability of the hypothesis HHH being true given the data DDD.

• P(D∣H)P(D | H)P(D∣H) is the likelihood: The probability of observing the data DDD given that hypothesis HHH is true.

• P(H)P(H)P(H) is the prior probability: The probability of the hypothesis before observing the data.

• P(D)P(D)P(D) is the marginal likelihood or evidence: The total probability of observing the data under all possible hypotheses.

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Conclusion

Probability distributions play a central role in statistics and probability theory, providing a framework to model uncertainty and randomness in various scenarios. Whether dealing with discrete or continuous random variables, understanding different types of probability distributions is essential for making informed decisions and predictions.

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Discrete distributions, such as the Binomial, Poisson, and Geometric distributions, are used to model countable events, like the number of successes in a series of trials or the occurrence of rare events in a fixed interval. On the other hand, continuous distributions, like the Normal, Exponential, and Uniform distributions, are used to model variables that can take an infinite number of possible values within a certain range, such as heights, weights, or the time between events.