Probability Distributions and their Stories

So far, we have looked a univariate distributions, but we will consider multivariate distributions in class, and you will encounter them in your research. First, we consider a discrete multivariate distribution, the Multinomial.

Multinomial distribution

• Story. This is a generalization of the Binomial distribution. Instead of a Bernoulli trial consisting of two outcomes, each trial has $K$ outcomes. The probability of getting $n_1$ of outcome 1, $n_2$ of outcome 2, ..., and $y_K$ of outcome $K$ out of a total of $N$ trials is Multinomially distributed.
• Example. There are two alleles in a population, A and a. Each individual may have genotype AA, Aa, or aa. The probability distribution describing having $y_1$ AA individuals, $y_2$ Aa individuals, and $n_3$ aa individuals in a population of $N$ total individuals is Multinomially distributed.
• Parameters. $N$, the total number of trials, and $θ={θ_1,θ_2,…θ_k}$, the probabilities of each outcome. Note that $∑_i θ_i=1$ and there is a further restriction that $∑_i y_i=N$.
• Support. The K-nomial distribution is supported on $\mathbb{N}^K$.
• Usage The usage below assumes that theta is a length K array.
  
  

  Package	Syntax
  NumPy	np.random.multinomial(N, theta)
  SciPy	scipy.stats.multinomial(N, theta)
  Stan sampling	multinomial(theta)
  Stan rng	multinomial_rng(theta, N)

  
  
• Probability density function.
  
  
  $\begin{align}f(\mathbf{y};\mathbf{\theta}, N) = \frac{N!}{y_1!\,y_2!\cdots y_k!}\,\theta_1^{y_1}\,\theta_2^{y_2}\cdots \theta_K^{y_K}\end{align}$
  
  
  
• Related distributions.
  • The Multinomial distribution generalizes the Binomial distribution to multiple dimensions.
• Notes.
  • For a sampling statement in Stan, the value of N is implied