Common Statistical Distributions

Understanding common statistical distributions is key in data science. These distributions help model real-world phenomena, guiding analysis and decision-making. From the normal distribution to the Poisson distribution, each serves unique purposes in statistical methods and mathematical modeling.

1. Normal (Gaussian) Distribution

   • Symmetrical, bell-shaped curve characterized by its mean (μ) and standard deviation (σ).
   • Central Limit Theorem states that the sum of a large number of independent random variables tends to be normally distributed.
   • Used in hypothesis testing and confidence intervals due to its properties.

2. Binomial Distribution

   • Models the number of successes in a fixed number of independent Bernoulli trials (n), each with the same probability of success (p).
   • Defined by two parameters: n (number of trials) and p (probability of success).
   • Useful for scenarios like coin flips or quality control in manufacturing.

3. Poisson Distribution

   • Describes the number of events occurring in a fixed interval of time or space, given a known average rate (λ).
   • Assumes events occur independently and at a constant rate.
   • Commonly used in fields like telecommunications and traffic flow analysis.

4. Exponential Distribution

   • Models the time between events in a Poisson process, characterized by the rate parameter (λ).
   • Memoryless property: the probability of an event occurring in the next time interval is independent of how much time has already elapsed.
   • Frequently applied in survival analysis and reliability engineering.

5. Uniform Distribution

   • All outcomes are equally likely within a specified range [a, b].
   • Defined by two parameters: minimum (a) and maximum (b).
   • Useful in simulations and scenarios where each outcome has the same probability.

6. Chi-Square Distribution

   • Used primarily in hypothesis testing and constructing confidence intervals for variance.
   • Defined by degrees of freedom, which typically correspond to the number of independent standard normal variables squared.
   • Commonly applied in goodness-of-fit tests and tests of independence.

7. Student's t-Distribution

   • Similar to the normal distribution but with heavier tails, making it more suitable for small sample sizes.
   • Defined by degrees of freedom, which affect the shape of the distribution.
   • Used in hypothesis testing and constructing confidence intervals when the population standard deviation is unknown.

8. F-Distribution

   • Used primarily in analysis of variance (ANOVA) and regression analysis.
   • Defined by two sets of degrees of freedom: one for the numerator and one for the denominator.
   • Helps compare variances between two populations.

9. Beta Distribution

   • Defined on the interval [0, 1] and characterized by two shape parameters (α and β).
   • Flexible in modeling random variables that are constrained within a finite range.
   • Commonly used in Bayesian statistics and modeling proportions.

10. Gamma Distribution

    • Generalizes the exponential distribution and is defined by a shape parameter (k) and a scale parameter (θ).
    • Models waiting times and is useful in queuing models and reliability analysis.
    • Includes the exponential and chi-square distributions as special cases.

11. Bernoulli Distribution

    • Represents a single trial with two possible outcomes: success (1) or failure (0).
    • Defined by a single parameter (p), the probability of success.
    • Fundamental building block for more complex distributions like the binomial distribution.

12. Geometric Distribution

    • Models the number of trials until the first success in a series of independent Bernoulli trials.
    • Defined by the probability of success (p) on each trial.
    • Useful in scenarios like determining the number of attempts needed to achieve a goal.

13. Negative Binomial Distribution

    • Generalizes the geometric distribution to model the number of trials needed to achieve a fixed number of successes.
    • Defined by the number of successes (r) and the probability of success (p).
    • Applicable in over-dispersed count data scenarios.

14. Lognormal Distribution

    • Models variables whose logarithm is normally distributed, resulting in a right-skewed distribution.
    • Commonly used in financial modeling and environmental data.
    • Useful for modeling non-negative variables that cannot be negative.

15. Weibull Distribution

    • Versatile distribution used in reliability analysis and life data analysis.
    • Defined by a shape parameter (k) and a scale parameter (λ), affecting its failure rate.
    • Can model increasing, constant, or decreasing failure rates depending on the value of k.