Probability Distribution Types

Understanding different types of probability distributions is key for making informed business decisions. Each distribution helps model various scenarios, from equal chances in uniform distribution to predicting rare events with Poisson distribution, guiding effective strategies.

1. Uniform Distribution

   • All outcomes are equally likely within a defined range.
   • Defined by two parameters: minimum (a) and maximum (b).
   • Useful in scenarios where each outcome has the same probability, such as rolling a fair die.

2. Bernoulli Distribution

   • Represents a single trial with two possible outcomes: success (1) or failure (0).
   • Defined by a single parameter, p, which is the probability of success.
   • Fundamental for understanding binary outcomes in business decisions.

3. Binomial Distribution

   • Models the number of successes in a fixed number of independent Bernoulli trials.
   • Defined by two parameters: n (number of trials) and p (probability of success).
   • Commonly used in quality control and market research to assess success rates.

4. Poisson Distribution

   • Models the number of events occurring in a fixed interval of time or space.
   • Defined by a single parameter, λ (lambda), which is the average rate of occurrence.
   • Useful for predicting rare events, such as customer arrivals or system failures.

5. Normal Distribution

   • Symmetrical, bell-shaped distribution characterized by its mean (μ) and standard deviation (σ).
   • Many natural phenomena and business metrics (e.g., heights, test scores) follow this distribution.
   • Central to the Central Limit Theorem, which states that the sum of many independent random variables tends toward a normal distribution.

6. Exponential Distribution

   • Models the time until an event occurs, such as waiting times or service times.
   • Defined by a single parameter, λ (rate), which is the inverse of the mean.
   • Commonly used in queuing theory and reliability analysis.

7. Beta Distribution

   • Defined on the interval [0, 1], useful for modeling random variables that represent proportions.
   • Characterized by two shape parameters, α (alpha) and β (beta), which determine the distribution's shape.
   • Often used in Bayesian statistics and project management for estimating probabilities.

8. Gamma Distribution

   • Generalizes the exponential distribution and models the time until the k-th event occurs.
   • Defined by two parameters: shape (k) and scale (θ).
   • Useful in various fields, including finance and insurance, for modeling waiting times.

9. Geometric Distribution

   • Models the number of trials until the first success in a series of independent Bernoulli trials.
   • Defined by a single parameter, p, which is the probability of success on each trial.
   • Applicable in scenarios like customer retention and marketing campaigns.

10. Hypergeometric Distribution

    • Models the number of successes in a sample drawn without replacement from a finite population.
    • Defined by three parameters: population size (N), number of successes in the population (K), and sample size (n).
    • Useful in quality control and survey sampling where the population is limited.