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MULTINOMIAL

Inprobability theory, the multinomial distribution is a generalization of thebinomial distribution.

For nindependenttrials each of which leads to a success for exactly one of k categories, with each

category having a given fixed success probability, the multinomial distribution gives the probability of

any particular combination of numbers of successes for the various categories.

The binomial distribution is theprobability distributionof the number of successes for one of just two

categories in n independentBernoulli trials, with the same probability of success on each trial. In a

multinomial distribution, the analog of the Bernoulli distribution is thecategorical distribution, where

each trial results in exactly one of some fixed finite number k of possible outcomes, with

probabilities p1, ..., pk (so that pi 0 for i = 1, ..., k and ), and there are n independent

trials. Then if the random variables Xi indicate the number of times outcome number i is observed over

the n trials, the vector X = (X1, ..., Xk) follows a multinomial distribution with parameters n and p,

where p = (p1, ..., pk).

Note that, in some fields, such asnatural language processing, the categorical and multinomial

distributions areconflated, and it is common to speak of a "multinomial distribution" when acategorical

distributionis actually meant. This stems from the fact that it is sometimes convenient to express the

outcome of a categorical distribution as a "1-of-K" vector (a vector with one element containing a 1 and

all other elements containing a 0) rather than as an integer in the range ; in this form, a

categorical distribution is equivalent to a multinomial distribution over a single observation.

POLYNOMIAL

Inmathematics, polynomials are the simplest class ofmathematical expressions(apart from the

numbers and expressions representing numbers). Apolynomial is an expression constructed

fromvariables(also calledindeterminates) andconstants(usually numbers, but not always), using only

the operations ofaddition,subtraction,multiplication, and non-negativeintegerexponents(which are

abbreviations for several multiplications by the same value). However, the division by a constant is

allowed, because themultiplicative inverseof a non-zero constant is also a constant. For

example, x2 x/4 + 7 is a polynomial, but x2 4/x + 7x3/2 is analgebraic expressionthat is not a

polynomial, because its secondterminvolves a division by the variable x (the term 4/x), and also

because its third term contains an exponent that is not a non-negative integer (3/2).

A polynomial function is a function which is defined by a polynomial. Sometimes, the term polynomial is

reserved for the polynomials that are explicitly written as a sum (or difference) of terms involving only

multiplications and exponentiation by non negative integer exponents. In this context, the other

polynomials are called polynomial expressions. For example, is a polynomial expression that

represents the same thing as the polynomial The term "polynomial", as an

adjective, can also be used for quantities that can be expressed as a polynomial of some parameter, as

inpolynomial time, which is used incomputational complexity theory.

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