Step By Step Calculus » 8.6 - Multinomial and Hypergeometric Distributions

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Synopsis
Multinomial and Binomial Coefficients: The number of ways we can divide nn items into rr distinct subgroups of sizes n_{1},n_{2},\cdots ,n_{r}n_{1},n_{2},\cdots ,n_{r} is the multinomial coefficient
\displaystyle \binom{n}{n_{1},n_{2},\cdots ,n_{r}}= \begin{cases} \frac{n!}{n_{1}!n_{2}!\cdots n_{r}!} & n_{1}+n_{2}+\cdots +n_{r}=n \\ 0 & \textrm{otherwise} \end{cases} .
\displaystyle \binom{n}{n_{1},n_{2},\cdots ,n_{r}}= \begin{cases} \frac{n!}{n_{1}!n_{2}!\cdots n_{r}!} & n_{1}+n_{2}+\cdots +n_{r}=n \\ 0 & \textrm{otherwise} \end{cases} .
If r=2r=2, we have a binomial coefficient:
\displaystyle \binom{n}{k}= \begin{cases} \frac{n!}{n!(n-k)!} & k \in \{0,1,\dots,n\} \\ 0 & \textrm{otherwise} \end{cases} .
\displaystyle \binom{n}{k}= \begin{cases} \frac{n!}{n!(n-k)!} & k \in \{0,1,\dots,n\} \\ 0 & \textrm{otherwise} \end{cases} .
Hypergeometric Coefficient: We draw nn items without replacement from a set of NN items made up of two groups AA and BB of sizes mm and N-mN-m, respectively. The number of ways to draw kk of the mm from group AA is
\displaystyle \#(\displaystyle \#(draw kk from AA\displaystyle )\times\#(\displaystyle )\times\#(draw n-kn-k from BB\displaystyle ) = \binom{m}{k}\binom{N-m}{n-k}. \displaystyle ) = \binom{m}{k}\binom{N-m}{n-k}.
Multinomial Theorem: For any n\in\mathbb{N}n\in\mathbb{N} (natural numbers)
\displaystyle (x_{1}+x_{2}+\cdots +x_{r})^{n}= \sum \limits_{\substack{\left( n_{1},n_{2},\cdots ,n_{r}\right) \\ n_{1}+n_{2}+\cdots +n_{r}=n }} \binom{n}{n_{1},n_{2},\cdots ,n_{r}}x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{r}^{n_{r}} ,
\displaystyle (x_{1}+x_{2}+\cdots +x_{r})^{n}= \sum \limits_{\substack{\left( n_{1},n_{2},\cdots ,n_{r}\right) \\ n_{1}+n_{2}+\cdots +n_{r}=n }} \binom{n}{n_{1},n_{2},\cdots ,n_{r}}x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{r}^{n_{r}} ,
where the summation is over all distinct vectors of non-negative integers that sum nn.
Binomial Theorem: If r=2r=2, we have the binomial theorem:
\displaystyle (x+y)^{n}=\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}.
\displaystyle (x+y)^{n}=\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}.
Terms in multi-index summation: It follows by properties of the binomial coefficient that
\displaystyle \sum_{\substack{n_1,n_2,\dots,n_r = 1 \\ n_1 + n_2 + \dots +n_r = n}}^{n} 1 = \binom{n-1}{r-1}.
\displaystyle \sum_{\substack{n_1,n_2,\dots,n_r = 1 \\ n_1 + n_2 + \dots +n_r = n}}^{n} 1 = \binom{n-1}{r-1}.
Hypergeometric Theorem: Suppose that n,m,Nn,m,N are non-negative integers. Then,
\displaystyle \binom{N}{n}=\sum_{k=0}^{n}\binom{m}{k}\binom{N-m}{n-k}.
\displaystyle \binom{N}{n}=\sum_{k=0}^{n}\binom{m}{k}\binom{N-m}{n-k}.
Multinomial and Binomial Probability Mass Function (pmf): Suppose the probability of drawing an item of type ii is p_ip_i for i=1,\cdots,ri=1,\cdots,r and p_1+\cdots+p_r=1p_1+\cdots+p_r=1. Then, the probability p(n_1, n_2, \dots, n_r)p(n_1, n_2, \dots, n_r) of having n_1n_1 objects of type 1, n_2n_2 objects of type 2, ..., and n_rn_r objects of type rr when drawing n=n_1 + n_2 + \dots + n_rn=n_1 + n_2 + \dots + n_r items with replacement is
\displaystyle p(n_1, n_2, \dots, n_r) = \binom{n}{n_{1},n_{2},...,n_{r}}p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{r}^{n_{r}}.
\displaystyle p(n_1, n_2, \dots, n_r) = \binom{n}{n_{1},n_{2},...,n_{r}}p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{r}^{n_{r}}.
If r=2r=2, then, letting p_1=pp_1=p and p_1=1-pp_1=1-p, we have the binomial pmf
\displaystyle p(k) = \binom{n}{k} p^{k}(1-p)^{n-k}
\displaystyle p(k) = \binom{n}{k} p^{k}(1-p)^{n-k}
Hypergeometric pmf: is the probability p(k)p(k) of drawing kk items of type AA in nn draws without replacement from the set of mm group AA items and N-mN-m group BB items. Then,
\displaystyle p(k) = \frac{\binom{m}{k}\binom{N-m}{n-k}}{\binom{N}{n}} \quad \displaystyle p(k) = \frac{\binom{m}{k}\binom{N-m}{n-k}}{\binom{N}{n}} \quad for \displaystyle k=0,1,\dots,m. \displaystyle k=0,1,\dots,m.