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Discrete Probability Distributions
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Binomial Distribution
Formula: where is the number of trials, is the number of successes, and is the probability of success on a single trial. Key Properties: Fixed number of trials, only two possible outcomes (success or failure), each trial is independent.
Poisson Distribution
Formula: where is the mean number of successes that occur in a specified region. Key Properties: Models the number of events in a fixed interval of time or space, the events occur with a known constant mean rate, the events occur independently.
Geometric Distribution
Formula: where is the number of trials until the first success and is the probability of success on each trial. Key Properties: Models the number of trials until the first success, each trial is independent, constant probability of success.
Negative Binomial Distribution
Formula: where is the total number of trials, is the number of successes, and is the probability of success on each trial. Key Properties: Generalizes the geometric distribution, trials are independent, counts the number of failures before a fixed number of successes is reached.
Hypergeometric Distribution
Formula: where is the population size, is the number of success states in the population, is the number of draws, and is the number of observed successes. Key Properties: Without replacement, fixed population size, each draw changes the probability of the next success.
Uniform Distribution
Formula: for where is the number of equally likely outcomes. Key Properties: All outcomes are equally likely, simple model for a fair random process, used for non-sequential events.
Bernoulli Distribution
Formula: for where is the probability of success (k=1). Key Properties: Simplest case of the binomial distribution (n=1), only two possible outcomes (success or failure), models a single trial.
Multinomial Distribution
Formula: , where is the number of trials and is the success probability of outcome . Key Properties: Generalizes the binomial distribution, models the outcome of trials where each trial can result in one of more than two categories.
Zipf's Distribution
Formula: for and . Key Properties: Models phenomena where frequency of an item is inversely proportional to its rank, often found in natural languages and city populations, characterized by the parameter which is called the exponent.
Logarithmic Series Distribution
Formula: for and . Key Properties: Models the number of occurrences within a unit time or space when the occurrences are very rare, each occurrence is independent, and used in ecological studies and informatics.
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