Multinomial distribution

Nature For an experiment in which successive outcomes can be classified into two or more categories and the probabilities associated with the respective outcomes remain constant, then the experiment can be characterized through the multinomial distribution. 

Any data set that consists of discrete classification into outcomes or descriptors is treated with a binomial (two outcomes) or multinomial (tliree or more outcomes) likelihood function. For example, if we have y successes from n experiments, e.g., y heads from n tosses of a coin or y green balls from a barrel filled with red and green balls in unknown proportions, the likelihood function is a binomial distribution ... 

If there are more than two outcomes, we can use the multinomial distribution for the likelihood ... 

Show that for this distribution uncorrelated and independent are equivalent Exercise. A molecule can occupy different levels nl,n2,... with probabilities P1,P2 Suppose there are N such molecules. The probability for finding the successive levels occupied by Nl,N2,... molecules is given by the multinomial distribution... 

The total number of coins C (20 in this case) serves as the constraint. The number of ways each distribution can be generated, which we will call Q, turns out to be given by what is called the multinomial expansion ... 

Just as with the binomial distribution, calculating factorials is tedious for large N. The binomial distribution converged to a Gaussian for large N (Equation 4.11). The most probable distribution for the multinomial expansion converges to an exponential ... 

Let n0 be an m-dimensional deterministic vector representing the number of particles contained in the drug amount qo initially given in each compartment. Also, let (t) be an m-dimensional random vector that takes on zero and positive integer values. (t) represents, at time t, the random distribution among the m compartments of the number of molecules starting in i. Since all of the molecules are independent by assumption, (f) follows a multinomial distribution ... 

When drugs are given in repeated dosage, we have to compile the repeated schemes. We assume linearity in mixing multinomial distributions, i.e., if... 

An important generalization concerns the multinomial distribution of observations at different times. To deal with this, we analyze in the Markovian context the prediction of the statistical behavior of particles at time t + t° based on the observations at t, i.e., the state about the conditional random variable [KAt + t°) ni (t)]. As previously, in common use is the multinomial distribution... 

Multinomial Probability distribution of the number of failures in n independent demands in which at each trial tliere are more than two possible outcomes Appropriate for situations similar to those for binomial distribution, except more than two outcomes can be found... 

By using the multinomial distribution, mathematical approximations, and simplifying assumptions, Gy (1992, p. 362) has derived an approximate formula for the variance of the FE (Var(FE)) ... 

In the previous sections our discussions has concentrated on just two r.v. s. We refer to their joint distribution as a bivariate distribution. When more than two r.v. s, say Xi, X2,..., Xk, are jointly distributed, we can similarly define their joint p.m.f. (or p.d.f.), denoted by/(xi, X2,..., x ), referred to as a multivariate distribution. The properties of a multivariate distribution are namral extensions of the bivariate distribution properties. Examples of a multivariate distribution are multinomial, multivariate, normal, and Dirichlet distributions. 

In this section, we introduce a multinomial distribution, an important family of discrete multivariate distributions. This family generalized the binomial family to the situation in which each trial has n (rather than two) distinct possible outcomes. Then multivariate normal and Dirichlet distributions wUl be discussed. 

Solution A r.v. vector (Xi.Xs) has a multinomial distribution with w = 10 trials,... 

Note that it is often convenient to maximize the log-likelihood function, a monotone function of the likelihood whose maximum will correspond to the maximum of the original likelihood function. If our variables are discrete, we instead use the multinomial distribution whose probability mass function is as follows ... 

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