Continuous Distributions

Overview

Continuous distributions describe variables that can take any value within a range (e.g., time, weight, distance, temperature). Unlike discrete variables, the probability of any single exact value is zero; probabilities are defined over intervals.

Common Distributions

• NORM: Normal (Gaussian) distribution. The most important distribution in statistics due to the Central Limit Theorem. Symmetric, bell-shaped.
• T_DIST: Student’s t-distribution. Similar to the Normal but with heavier tails. Used when sample sizes are small.
• UNIFORM: Uniform distribution. All outcomes in an interval are equally likely.
• EXPON: Exponential distribution. Models the time between independent events (memoryless property).
• BETA: Beta distribution. Defined on the interval [0, 1]. Often used to model probabilities or proportions.
• WEIBULL_MIN: Weibull distribution. Widely used in reliability engineering and survival (time-to-failure) analysis.

Figure 1: Beta Distribution Shapes: The Beta distribution is highly flexible. Depending on parameters alpha and beta, it can be U-shaped (0.5, 0.5), uniform (1, 1), or bell-shaped (2, 2).

Specialized Distributions

• CAUCHY: Heavy-tailed distribution with undefined mean and variance. Known in physics as the Lorentz distribution.
• LOGNORM: Log-normal distribution. Used for variables that are products of independent random variables (e.g., stock prices, income).
• PARETO: Power-law distribution. Models wealth distribution (“80-20 rule”) and other unequal phenomena.
• LAPLACE: Double exponential distribution. Used in regression (Lasso) and signal processing.

Native Excel Capabilities

Excel covers the basics well (NORM.DIST, BETA.DIST, LOGNORM.DIST, WEIBULL.DIST). However, for distributions like Cauchy, Laplace, Pareto, or Levy, users often have to manually construct formulas using standard mathematical functions, which creates friction and opportunity for error.

Tools

Tool	Description
BETA	Wrapper for scipy.stats.beta distribution providing multiple statistical methods.
CAUCHY	Wrapper for scipy.stats.cauchy distribution providing multiple statistical methods.
CHISQ	Compute various statistics and functions for the chi-squared distribution from scipy.stats.chi2.
EXPON	Exponential distribution function wrapping scipy.stats.expon.
F_DIST	Unified interface to the main methods of the F-distribution, including PDF, CDF, inverse CDF, survival function, and distribution statistics.
LAPLACE	Laplace distribution function supporting multiple methods.
LOGNORM	Compute lognormal distribution statistics and evaluations.
NORM	Normal (Gaussian) distribution function supporting multiple methods.
PARETO	Generalized Pareto distribution function supporting multiple methods.
T_DIST	Student’s t distribution function supporting multiple methods from scipy.stats.t.
UNIFORM	Uniform distribution function supporting multiple methods.
WEIBULL_MIN	Compute various functions of the Weibull minimum distribution using scipy.stats.weibull_min.