Geometric & Negative Binomial Distributions

• Geometric distribution X ~ Geo(p): models the number of trials needed to get the first success.
• Four conditions must hold: indefinite trials, independence, two outcomes per trial, constant probability of success.
• PMF: P(X = x) = p(1−p)^x−1, for x = 1, 2, 3, …
• CDF: P(X ≤ x) = 1 − (1−p)^x, and P(X > x) = (1−p)^x
• Mean E(X) = 1/p, Variance Var(X) = (1−p)/p²
• The Geometric distribution has no memory — past failures don't affect future probabilities.
• The mode is always X = 1 (probabilities strictly decrease from the first trial onward).
• Negative Binomial X ~ Negative B(r, p): models the number of trials needed to get a fixed number, r, of successes.
• Same four conditions as Geometric, but now r successes are required instead of just 1.
• PMF: P(X = x) = ^x−1C_r−1 · p^r(1−p)^x−r, for x = r, r+1, r+2, …
• Mean E(X) = r/p, Variance Var(X) = r(1−p)/p²
• When r = 1, the Negative Binomial formula collapses exactly into the Geometric formula.

Picture rolling a die and waiting for your first six. You don't know if it'll take 1 roll, 5 rolls, or 50 rolls — but eventually (with probability 1) it will happen. The Geometric distribution is the tool for exactly this situation: counting trials until the very first success happens.

We write X ~ Geo(p), where:

• X = the number of trials it takes to get the first success (so X = 8 means the first success happened on the 8th go, meaning the first 7 were failures)
• p = the fixed probability of success on any single trial

A Geometric model is only valid if all four of these hold:

1. The experiment consists of an indefinite number of successive trials
2. The outcome of each trial is independent of all the others
3. Each trial has exactly two possible outcomes (success / failure)
4. The probability of success is constant, p, on every trial

geometric sequence decreasing