Probability Distributions

Math for CS · Axiom Academy

PMF, CDF, expected value, and the distribution zoo A probability distribution is a complete mathematical description of a random phenomenon. Rather than listing individual outcomes, a distribution gives us a formula that covers every possible value a random variable can take. Once you identify which distribution applies, you instantly know its mean, variance, and behavior. Probability Mass Function (PMF) For a discrete random variable X , the PMF assigns a probability to each possible value: A valid PMF satisfies two properties: Cumulative Distribution Function (CDF) The CDF gives the probability that X is at most some value: The average squared deviation from the mean: Single trial: success (1) with probability p , failure (0) with probability 1-p . Number of successes in n independent Bernoulli trials. Number of trials until the first success. Count of events in a fixed interval at average rate . Equal probability over interval [a,b] . PDF: f(x) = 1/(b-a) . Time until next event in a Poisson process. PDF: . Choosing the Right Distribution A distribution is fully specified by its PMF/PDF and parameters PMF gives point probabilities; CDF gives cumulative probabilities E[X] and summarize center and spread Matching the right distribution to your problem is the crucial skill Discrete: Bernoulli, Binomial, Geometric, Poisson Continuous: Uniform, Normal, Exponential Axiom Academy | Probability for Computer Science

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