Random Variables

Checkpoints

• Bernoulli Distribution Essential The Bernoulli distribution models a single binary trial: X = 1 with probability p and X = 0 with probability 1 − p. This checkpoint defines the distribution, computes its mean and variance, and frames it as the atomic building block from which Binomial, Geometric, and many other discrete distributions are constructed.
• Binomial Distribution Essential The Binomial distribution counts the number of successes in n independent Bernoulli(p) trials, with PMF P(X = k) = C(n,k) p^k (1−p)^(n−k). This checkpoint derives the PMF from the Bernoulli building blocks, computes the mean np and variance np(1−p), and exhibits the additive structure: the sum of independent Binomials with the same p is again Binomial.
• Random Variables Essential A random variable is a measurable function from a probability space (Ω, ℱ, P) to (ℝ, ℬ(ℝ)); its push-forward measure is the distribution. This checkpoint develops the definition, distinguishes discrete from absolutely continuous random variables, introduces the cumulative distribution function (CDF), probability mass function (PMF), and probability density function (PDF), and explains why measurability is exactly the condition that makes probabilities of events about the variable well-defined.
• Exponential Distribution Essential The Exponential distribution with rate λ has density f(x) = λ e^(−λx) for x ≥ 0 and is the unique absolutely continuous distribution on [0, ∞) with the memorylessness property. This checkpoint constructs the distribution, computes its mean 1/λ and variance 1/λ², and identifies it as the continuous-time analogue of the Geometric distribution and the inter-arrival law of a Poisson process.
• Gamma Distribution Essential The Gamma distribution with shape α > 0 and rate λ > 0 has density f(x) = λ^α x^(α−1) e^(−λx) / Γ(α) on (0, ∞), where Γ is the gamma function. This checkpoint defines the distribution, shows that a sum of α independent Exponential(λ) random variables is Gamma(α, λ) for integer α, and works through the mean α/λ and variance α/λ².
• Geometric Distribution Essential The Geometric distribution models the number of independent Bernoulli(p) trials needed to obtain the first success. This checkpoint derives the PMF P(X = k) = (1−p)^(k−1) p, computes the mean 1/p and variance (1−p)/p², and proves the memorylessness property that uniquely characterises the Geometric among discrete distributions on the positive integers.
• Normal Distribution Essential The Normal (Gaussian) distribution N(μ, σ²) has density f(x) = (2πσ²)^(−1/2) exp(−(x − μ)² / 2σ²) on ℝ. This checkpoint defines the standard and general Normal, verifies that the density integrates to one (the classical √(2π) computation), computes its mean μ and variance σ², and explains why the Normal is the limiting distribution of standardised sums via the Central Limit Theorem.
• Poisson Distribution Essential The Poisson distribution with rate λ assigns probability P(X = k) = e^(−λ) λ^k / k! and arises as the limit of Binomial(n, λ/n) as n → ∞. This checkpoint takes that limit explicitly, derives the mean and variance (both equal to λ), and motivates the Poisson as the canonical model for counts of rare independent events in a fixed window.
• Relationships Between Common Distributions Essential The standard distributions are not independent objects; they sit inside a tight web of structural and limiting relationships. This checkpoint maps out both kinds: structurally, Bernoulli is the n = 1 Binomial, the Geometric is the discrete analogue of the Exponential, and the Exponential is the α = 1 Gamma; in the limit, Binomial(n, λ/n) → Poisson, sums of Exponential(λ) are Gamma(α, λ), and standardised sums of any finite-variance distribution converge to Normal.