= Multinomial Distribution = The '''multinomial distribution''' is a discrete probability density function. <> ---- == Description == Where the [[Analysis/BinomialDistribution|binomial distribution]] requires two outcomes for any trial, the multinomial distribution allows for multiple exclusive and exhaustive outcomes. There are ''k'' possible outcomes, each with an associated probability of ''p,,i,,'' for outcome ''i''. The count of observations that realize into outcome ''i'' is a random variable, denoted ''X,,i,,''. The probability of any particular outcome is expressed as ''Prob(X,,1,,=x,,1,,``, X,,2,,=x,,2,,``, ... X,,k,,=x,,k,,)''. Further note that it is required for... * the sum of ''p,,i,,'' across all outcomes is ''1''. * the sum of ''x,,i,,'' across all outcomes is ''n''. {{attachment:multi.svg}} ---- == Moments == It doesn't necessarily make sense to consider the [[Analysis/ExpectedValue|expected value]] of all multinomially distributed random variables at the same time. For any one of the ''X,,i,,'' random variables, the expected value is given as ''E[X,,i,,] = np,,i,,''. [[Analysis/Variance|Variance]] is given as ''Var[X,,i,,] = np,,i,,(1 - p,,i,,)''. These of course match the [[Analysis/BinomialDistribution|binomial distribution]]. The [[Analysis/Covariance|covariance matrix]] of a multinomial distribution is constructed as: * variances on the diagonal, as above * covariances calculated as ''Cov(X,,i,,``, X,,j,,) = -np,,i,,p,,j,,'' ---- CategoryRicottone