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Formula: $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$ where $\lambda$ 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.

Formula: $P(X=k) = p^k(1-p)^{1-k}$ for $k\in\{0,1\}$ where $p$ 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.

Formula: $P(X=k) = (1-p)^{k-1}p$ where $k$ is the number of trials until the first success and $p$ 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.

Formula: $P(X=k) = \frac{1}{n}$ for $k=1,2,...,n$ where $n$ 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.

Formula: $P(X_1=k_1,...,X_n=k_n) = \frac{n!}{k_1!...k_n!}p_1^{k_1}...p_n^{k_n}$, where $n$ is the number of trials and $p_i$ is the success probability of outcome $i$. Key Properties: Generalizes the binomial distribution, models the outcome of $n$ trials where each trial can result in one of more than two categories.

Formula: $P(X=k) = \frac{1/k^s}{\sum_{i=1}^N (1/i^s)}$ for $k=1,2,...,N$ and $s>0$. 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 $s$ which is called the exponent.

Formula: $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ where $n$ is the number of trials, $k$ is the number of successes, and $p$ 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.

Formula: $P(X=k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$ where $N$ is the population size, $K$ is the number of success states in the population, $n$ is the number of draws, and $k$ is the number of observed successes. Key Properties: Without replacement, fixed population size, each draw changes the probability of the next success.

Formula: $P(X=k) = -\frac{1}{\ln(1-p)} \cdot \frac{p^k}{k}$ for $k=1,2,...$ and $0 < p < 1$. 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.