Normal Distribution
| Item | Description |
|---|
PDF: f(x) = (1/(sigma*sqrt(2*pi))) * exp(-(x-mu)^2 / (2*sigma^2)) | Bell-shaped, symmetric around mean mu, spread sigma |
Notation: X ~ N(mu, sigma^2) | Mean mu, variance sigma^2 |
Standard Normal: Z ~ N(0,1) | Z = (X - mu) / sigma — subtract mean, divide by std dev |
68-95-99.7 Rule | 68% within 1 sigma, 95% within 2 sigma, 99.7% within 3 sigma |
Use Cases | Heights, test scores, measurement errors — any sum of many small effects |
Pitfall | Not suitable for skewed data; log-transform first if needed |
Binomial Distribution
| Item | Description |
|---|
PMF: P(X=k) = C(n,k) * p^k * (1-p)^(n-k) | Probability of exactly k successes in n independent trials |
Notation: X ~ Bin(n, p) | n = number of trials, p = probability of success per trial |
Mean = n*p, Variance = n*p*(1-p) | Expected successes and spread |
Use Cases | Coin flips, defect rates, A/B test conversions, survey responses |
Normal Approximation | When n*p >= 10 and n*(1-p) >= 10, Bin(n,p) ~= N(n*p, n*p*(1-p)) |
Pitfall | Trials must be independent; without replacement, use hypergeometric |
Poisson Distribution
| Item | Description |
|---|
PMF: P(X=k) = (lambda^k * e^(-lambda)) / k! | Probability of k events in a fixed interval |
Notation: X ~ Poisson(lambda) | lambda = mean rate (events per interval) |
Mean = Variance = lambda | Unique property — mean equals variance |
Use Cases | Website hits per hour, call center arrivals, rare disease incidence, typos per page |
Additive Property | Sum of independent Poissons is Poisson with lambda = sum of lambdas |
Pitfall | Events must be independent; clustered events violate assumption |
Exponential Distribution
| Item | Description |
|---|
PDF: f(x) = lambda * e^(-lambda*x) for x >= 0 | Models waiting time until first event in a Poisson process |
Notation: X ~ Exp(lambda) | lambda = rate parameter, 1/lambda = mean waiting time |
Mean = 1/lambda, Variance = 1/lambda^2 | Longer average wait = higher variance |
Memoryless Property | P(X > s+t | X > s) = P(X > t) — past waiting doesn't affect future |
Use Cases | Time between arrivals, equipment failure times, radioactive decay |
Pitfall | Constant failure rate assumption; use Weibull for changing rates |
Pro Tip: The normal distribution is the default assumption in most statistical tests. Always check if your data is actually normally distributed before applying Z-tests or t-tests.