Discrete Distributions
Discrete distributions describe outcomes that are distinct and countable. Heads or tails. Number of clicks. Number of customer support tickets in an hour. The value can only be a whole number — you can't have 2.7 clicks.
Bernoulli Distribution
Models a single binary trial with two possible outcomes: success (1) or failure (0). Single parameter p is the probability of success.
When it shows up: Did the user click the ad? Did the transaction succeed? Did the patient survive? Any yes/no outcome is Bernoulli.
Mean: p. Variance: p(1−p).
Binomial Distribution
Models the number of successes in n independent Bernoulli trials. Parameters: n (number of trials) and p (probability of success per trial).
When it shows up: Out of 10,000 marketing emails sent, how many will be opened? Out of 500 A/B test participants, how many convert? Any "count of successes in fixed trials" is binomial.
Mean: np. Variance: np(1−p).
Poisson Distribution
Models the number of events occurring in a fixed interval of time or space, given a constant average rate. Single parameter (the rate).
When it shows up: Customer service calls per hour. API requests per second. Defects per meter of manufacturing line. Any "count of independent events arriving at a steady rate" is Poisson.
Mean: . Variance: . (The mean and variance are equal — this is a unique property of the Poisson.)
Count of successes in n independent Bernoulli trials. As n increases or p approaches 0.5, the distribution becomes more symmetric and bell-shaped.
P(X=k) = C(n,k) p^k (1−p)^(n−k)
Toggle between Bernoulli, Binomial, and Poisson. Adjust parameters with the sliders and watch the PMF bars shift — hover any bar to see its exact probability.
You're modeling the number of fraudulent transactions per day on a payment platform, where fraud events arrive independently and at a roughly constant average rate. Which distribution best describes this count?