Discrete Distributions

By Victor Arce

The bell curve from the last article grew out of counting coin flips. Counting is exactly where the discrete distributions come from too — each one answers a different question about whole-number outcomes. Here are the four you'll meet most.

Step through them below; the bars are the probability of each outcome.

Four distributions, one building block

• Bernoulli — a single yes/no trial: success with probability p, failure with 1 − p. It's the atom every other distribution here is built from.
• Binomial — count the successes in n independent Bernoulli trials. This is the coin-flip histogram from the previous article; it peaks near n·p and, for large n, fills into the normal curve.
• Geometric — count the trials up to the first success. Each extra failure is less likely than the last, so the bars decay geometrically. It's the "how long until it finally works?" distribution.
• Poisson — count how many rare events land in a fixed window when the average rate is λ. Arrivals at a queue, typos per page, decays per second — all cluster around λ.

How they connect

These aren't four unrelated formulas — they're one idea seen from different angles. Stack Bernoulli trials and ask "how many successes?" → Binomial. Ask "how many trials until the first success?" → Geometric. Take a Binomial with huge n and tiny p (many chances, each unlikely) and it converges to Poisson. Recognizing which question you're asking is most of the work of choosing a distribution.

Back to Probability & Distributions