Multinomial Distribution

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The Multinomial Distribution: generalizes the Binomial Distribution to more than two possible outcomes. It models the probabilities of counts for multiple categories in $n$ independent trials, where each trial results in one of $k$ categories. If there are k categories, and in $n$ trials, you observe $x_1$ counts of category 1, $x_2$ counts of category 2, etc., the probability is given by:

$$ P(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k) = \frac{n!}{x_1! x_2! \dots x_k!} p_1^{x_1} p_2^{x_2} \dots p_k^{x_k} $$

Where $p_i$ is the probability of category $i$ and $x_1 + x_2 + \ldots + x_k = n$.

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Example: Rolling a 6-sided die $n$ times and counting how many times each face appears follows a multinomial distribution.
Relation to Binomial: When $k = 2$, the multinomial distribution reduces to the binomial distribution.

Hypergeometric Distribution

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The Hypergeometric Distribution: describes the probability of drawing a specific number of successes (without replacement) from a finite population that contains a fixed number of successes and failures. The probability of getting $k$ success in $n$ draws from a population of $N$ items containing $K$ successes is given by

$$ P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{\binom{N}{n}} $$

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Example: If you have a deck of 52 cards (with 26 red cards), the probability of drawing exactly 5 red cards when drawing 10 cards without replacement follows a hypergeometric distribution.

Relation to the Binomial Distribution:

Recall:

The Binomial Distribution models the probability of $k$ successes in $n$ independent trials with replacement (or when the population size is infinite).

When the population size $N$ is large enough, the hypergeometric distribution approximates the binomial distribution.

Key Difference: The key distinction is whether the trials are with replacement (binomial) or without replacement (hypergeometric).

Cauchy Distribution

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The Cauchy Distribution: a continuous probability distribution with undefined mean and variance. It has "heavy tails," meaning that extreme values have a non-negligible probability. The PDF is given by

$$ f(x) = \frac{1}{\pi (1 + x^2)} $$

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Example: The classic example of the Cauchy distribution comes from the intersection of two independent normally distributed random variables, where the ratio of the two follows a Cauchy distribution.
Properties:
- The Cauchy distribution has no defined mean, variance, or higher moments. Its median and mode are both 0.