Probability Distributions

A probability distribution assigns a probability to each possible outcome of a random variable. Random variables are either discrete (countable outcomes, described by a probability mass function PMF with P(X = k) and Σ P(X = k) = 1) or continuous (uncountable outcomes, described by a probability density function PDF with f(x) ≥ 0 and ∫ f(x) dx = 1). The cumulative distribution function CDF, F(x) = P(X ≤ x), applies to both types and is used to compute interval probabilities: P(a < X ≤ b) = F(b) - F(a). For a continuous variable, P(X = x) = 0 for any single point, so probability is associated with intervals, not individual values. Discrete distributions. The binomial distribution gives the probability of k successes in n independent Bernoulli trials with constant success probability p: P(X = k) = C(n,k) p^k (1-p)^(n-k), with mean np and variance np(1-p). The Poisson distribution models the number of events in a fixed interval when events occur independently at a constant average rate λ: P(X = k) = λ^k e^(-λ) / k!, with mean and variance both equal to λ. It is the limiting case of the binomial when n is large and p is small with np = λ. The geometric distribution gives the number of trials until the first success, with P(X = k) = (1-p)^(k-1) p, mean 1/p, and variance (1-p)/p². Continuous distributions. The uniform distribution on [a, b] has constant density 1/(b-a), mean (a+b)/2, and variance (b-a)²/12. The exponential distribution models the time between events in a Poisson process, with PDF f(x) = λ e^(-λx) for x ≥ 0, mean 1/λ, and variance 1/λ². It is memoryless: P(X > s + t | X > s) = P(X > t). The normal (Gaussian) distribution has PDF f(x) = (1/(σ√(2π))) exp(-(x-μ)²/(2σ²)), mean μ, and variance σ²; the standard normal has μ = 0 and σ = 1. By the central limit theorem, sums and averages of many independent random variables tend toward normality regardless of the underlying distribution. The Weibull distribution, with shape parameter k and scale parameter λ, is widely used in reliability engineering to model time to failure; k < 1 indicates decreasing failure rate (infant mortality), k = 1 gives the exponential distribution (constant failure rate), and k > 1 indicates increasing failure rate (wear-out). Choosing the right distribution depends on the physical mechanism, the nature of the data (discrete vs. continuous), and whether the process is memoryless or has a defined failure pattern.