Random Variables

Binomial(n, p)

models # of successes in n independent trials with chance p of success.

Outcomes: 0, 1, …n

Moments: 𝜇 = 𝑛𝑝, 𝜎² = 𝑛𝑝(1 − 𝑝)

PMF: Pr[𝑋 = 𝑘] = (𝑛/𝑘)𝑝^𝑘(1 − 𝑝)^𝑛−𝑘

Geometric(p)

models # of trials before first success, each trial independent w/ chance p of success

Outcomes: 0, 1, 2, …

Moments: 𝜇 = (1 − 𝑝)/𝑝, 𝜎² = (1 − 𝑝)/𝑝²

PMF: Pr[𝑋 = 𝑘] = 𝑝(1 − 𝑝)^𝑘

CDF: Pr[𝑋 ≤ 𝑘] = 1 − (1 − 𝑝)^⌊𝑘⌋+1

Poisson(𝜆)

models count of occurrences when events occur independently with average rate 𝜆

Outcomes: 0, 1, 2, …

Moments: 𝜇 = 𝜎2 = 𝜆

PMF: Pr[𝑋 = 𝑘] = 𝜆^𝑘𝑒^−𝜆/𝑘!

Discrete Uniform(a, b)

models n-sided fair die with numbers a, a + 1, …, b on it, 𝑛 = 𝑏 − 𝑎 + 1

outcomes: 𝑎, 𝑎+1, 𝑏

moments: 𝜇 = (𝑎 + 𝑏)/2, 𝜎² = (𝑛2 − 1)/12

PMF: Pr[𝑋 = 𝑘] = 1/𝑛

Normal(𝜇, 𝜎²)

models most things that that are the sum/mean of many smaller independent componants

outcomes: (−∞, ∞),

moments: 𝜇 = 𝜇, 𝜎² = 𝜎²

PMF: 𝑓(𝑥) = 1 / 𝜎√2𝜋 𝑒^{− 12 ( 𝑥−𝜇 / 𝜎 )2}

Continuous Uniform (a, b)

models things when anything [a, b] is equally likely

outcomes: [𝑎, 𝑏]

moments: 𝜇 = (𝑎 + 𝑏)/2, 𝜎² = (𝑏 − 𝑎)²/12

PDF: 𝑓(𝑥) = 1/(𝑏 − 𝑎)
CDF: 𝐹 (𝑥) = (𝑥 − 𝑎)/(𝑏 − 𝑎)

Exponential(𝜆)

continuous version of geometric, models wait time between possion events

outcomes: [0, ∞)

moments: 𝜇 = 1/𝜆, 𝜎² = 1/𝜆²

PDF: 𝑓(𝑥) = 𝜆𝑒^−𝜆𝑥
CDF: 𝐹 (𝑥) = 1 − 𝑒^−𝜆𝑥