Maria Cristina Mariani - Data Science in Theory and Practice
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- Название:Data Science in Theory and Practice
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Data Science in Theory and Practice: краткое содержание, описание и аннотация
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will also earn a place in the libraries of practicing data scientists, data and business analysts, and statisticians in the private sector, government, and academia.
, the multinomial probabilities are the coefficients in the multinomial expansion
, so they sum to 1. This expansion in fact gives the name of the distribution.
as a success and everything else a failure, then
simply counts successes in
independent trials and thus
. Thus, the first moment of the random vector and the diagonal elements in the covariance matrix are easy to calculate as
and
, respectively. The off‐diagonal elements (covariances) are not that complicated to calculate either. However, for multinomial random vectors, the first two moments are difficult to compute. The one‐dimensional marginal distributions are binomial; however, the joint distribution of
, the first
components, is not multinomial. Instead, suppose we group the first
categories into 1 and we let
. Because the categories are linked, that is,
, we also have that
. We can easily verify that the vector
, or equivalently
, will have a multinomial distribution with associated probabilities
.
components given the last
components. That is, the distribution of
and probabilities
, where
.
is said to have a
‐dimensional multivariate normal distribution (denoted
, where
is
‐dimensional multivariate normal distribution) with mean vector
and covariance matrix
if its density can be written as
may have any elements in
, but, just as in the one‐dimensional case, the standard deviation has to be positive. In the multivariate case, the covariance matrix
has to be symmetric and positive definite.