Types of Distributions in Probability

🎲 Probaility Mass Function (Discrete Data)

Used when outcomes are countable — like coin flips, number of visitors, or email opens.

Distribution	Variable Type	Shape / Nature	Typical Use Case	Data Science Example
Bernoulli	Binary (0/1)	Two outcomes only	Success/Failure	Spam (1) or Not Spam (0)
Binomial	Count of successes in n trials	Symmetric for p=0.5	Series of coin flips	A/B test results
Poisson	Count of events in time or space	Skewed right	Number of arrivals	Website hits/hour
Geometric	Trials until first success	Exponentially decreasing	Waiting for first event	First customer purchase
Negative Binomial	Trials until r-th success	Similar to geometric	Repeat experiments	Marketing conversions
Multinomial	Categorical outcomes (>2)	Multiple discrete bars	Multi-class probabilities	NLP topic modeling
Hypergeometric	Sampling without replacement	Finite population	Defects in batch	Quality control

🧩 PMF Graph Look: Bars or spikes — each bar = probability of one discrete value.

Used when values are measurable and continuous — like height, time, or temperature

Distribution	Variable Type	Shape / Nature	Typical Use Case	Data Science Example
Normal (Gaussian)	Continuous	Bell curve, symmetric	Natural phenomena	Regression errors, GMM
Uniform	Continuous	Flat line	Equal probability	Random sampling, simulation
Exponential	Continuous (positive)	Rapid decay	Time between events	Time-to-failure prediction
Log-Normal	Continuous (positive)	Right-skewed	Positive-only variables	Income, transaction values
Gamma	Continuous (positive)	Right-skewed, flexible	Duration modeling	Reliability analysis
Beta	Continuous (0–1)	Bounded (0–1)	Probabilities, ratios	Bayesian modeling
Chi-Square	Continuous (positive)	Right-skewed	Variance modeling	Hypothesis testing
Weibull	Continuous (positive)	Variable skew	Lifetimes, failure	Survival analysis
Pareto	Continuous (positive)	Heavy-tailed	Inequality, extremes	Customer lifetime value
Cauchy / t-distribution	Continuous	Heavy tails	Outlier-robust modeling	Bayesian inference

🧩 PDF Graph Look: Smooth curve — area under curve = total probability = 1.

Category	Used For	Key Examples	Graph Shape
PMF	Discrete probability	Bernoulli, Poisson, Binomial	Spikes / Bars
PDF	Continuous probability	Normal, Exponential, Beta	Smooth curve

Model Type	Underlying Distribution	Function Used
Classification (binary/multi-class)	Bernoulli, Multinomial	PMF
Regression	Normal (residuals)	PDF
Clustering	Gaussian (each cluster)	PDF
Anomaly Detection	Gaussian, Pareto	PDF/CDF
Reliability / Survival	Weibull, Gamma	PDF + CDF
Logistic Regression	Logistic	CDF
Bayesian Inference	Beta, Normal	PDF