Discrete Distributions Summary

Probability · Axiom Academy

SUMMARY Discrete Distributions Let's review the key discrete probability distributions and when to use each one. Binomial: Counts successes in a fixed number of independent trials with constant probability Geometric: Counts trials needed to get the first success in independent trials Negative Binomial: Counts trials needed to get the r-th success Bernoulli: Single yes/no trial (special case of Binomial with n=1) Hypergeometric: Like Binomial but sampling without replacement from finite population Discrete Uniform: All outcomes equally likely (fair die, random selection) Poisson: Models rare events occurring independently at a constant average rate over time or space Key Property: Mean equals variance ( ) Applications: Phone calls per hour, typos per page, accidents per month Decision Flowchart: Choosing the Right Distribution Binomial vs. Hypergeometric: Use hypergeometric when sampling without replacement from a small population Independence: Binomial requires independent trials; if outcomes affect each other, it's not binomial Geometric Support: Remember geometric starts at k=1 (first trial could be success), not k=0 Poisson Approximation: Can approximate Binomial when n is large and p is small (np stays moderate) Parameter Confusion: Always check what the parameters represent (n, p, λ, etc.) Mean = Variance: Only true for Poisson; helps identify when Poisson is appropriate

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