Discrete Distributions
| Distribution | PMF | Mean | Variance | Use Case |
|---|
| Bernoulli(p) | p^x * (1-p)^(1-x) for x in {0,1} | p | p(1-p) | Single coin flip, binary outcome |
| Binomial(n,p) | C(n,x)*p^x*(1-p)^(n-x) | np | np(1-p) | Number of successes in n trials |
| Poisson(lambda) | lambda^x*e^(-lambda)/x! | lambda | lambda | Counts of rare events |
| Geometric(p) | (1-p)^(x-1)*p | 1/p | (1-p)/p^2 | Trials until first success |
| NegBinomial(r,p) | C(x-1,r-1)*p^r*(1-p)^(x-r) | r/p | r(1-p)/p^2 | Trials until r successes |
| Hypergeometric(N,K,n) | C(K,x)*C(N-K,n-x)/C(N,n) | n*K/N | n*K/N*(N-K)/N*(N-n)/(N-1) | Sampling without replacement |
| Discrete Uniform(a,b) | 1/n for n values | (a+b)/2 | (n^2-1)/12 | Fair die, random digit |
Continuous Distributions
| Distribution | PDF | Mean | Variance | Use Case |
|---|
| Normal(mu,sigma^2) | 1/(sigma*sqrt(2pi))*exp(-(x-mu)^2/(2sigma^2)) | mu | sigma^2 | Most common — CLT-based |
| Standard Normal N(0,1) | 1/sqrt(2pi)*exp(-x^2/2) | 0 | 1 | Z-scores, standardized values |
| Exponential(lambda) | lambda*exp(-lambda*x), x>=0 | 1/lambda | 1/lambda^2 | Waiting times (memoryless) |
| Gamma(alpha,beta) | beta^alpha*x^(alpha-1)*exp(-beta*x)/Gamma(alpha) | alpha/beta | alpha/beta^2 | Sum of alpha exponentials |
| Beta(alpha,beta) | x^(alpha-1)*(1-x)^(beta-1)/B(alpha,beta) on [0,1] | alpha/(alpha+beta) | (alpha*beta)/((a+b)^2*(a+b+1)) | Probabilities, proportions, success rates |
| Uniform(a,b) | 1/(b-a) on [a,b] | (a+b)/2 | (b-a)^2/12 | Random number generation, non-informative prior |
| LogNormal(mu,sigma^2) | See formula | exp(mu+sigma^2/2) | (exp(sigma^2)-1)*exp(2mu+sigma^2) | Stock prices, file sizes, income |
| Student t(df=nu) | See formula | 0 (nu>1) | nu/(nu-2) (nu>2) | Unknown variance, small samples |
Sampling Distributions
| Item | Description |
|---|
t-Distribution | Use for t-tests — fatter tails than normal, converges to normal as df increases |
Chi-Square(df=k) | Sum of k squared standard normals — used in chi-square tests, confidence for variance |
F-Distribution(df1,df2) | Ratio of two chi-squares — ANOVA, comparing variances |
Central Limit Theorem | Sample mean ~ Normal(mu, sigma^2/n) as n -> infinity — justification for much of statistics |
Choosing a Distribution
| Item | Description |
|---|
Count data | Poisson (rare/independent), Binomial (fixed trials), Negative Binomial (overdispersed) |
Continuous positive | Exponential (waiting), Gamma (more flexible), LogNormal (multiplicative processes) |
Proportions [0,1] | Beta distribution — very flexible shape on unit interval |
Times to failure | Exponential (constant hazard), Weibull (changing hazard), LogNormal |
Pro Tip: Choose distributions from the data-generating process, not the data's histogram. Binomial for counts of successes, Poisson for rare event counts, Exponential for waiting times.