Multinomial Distribution
Definition
The multinomial pdf is
where x = (x1,
... , xk)
gives the number of each of k outcomes in n trials
of a process with fixed probabilities p = (p1,
... , pk)
of individual outcomes in any one trial. The vector x has
non-negative integer components that sum to n.
The vector p has non-negative integer components
that sum to 1.
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Background
The multinomial distribution is a generalization of the binomial distribution.
The binomial distribution gives the probability of the number of "successes"
and "failures" in n independent
trials of a two-outcome process. The probability of "success"
and "failure" in any one trial is given by the fixed
probabilities p and q =
1–p. The multinomial distribution gives
the probability of each combination of outcomes in n independent
trials of a k-outcome process. The probability
of each outcome in any one trial is given by the fixed probabilities p1,
... , pk.
The expected value of outcome i is npi.
The variance of outcome i is npi(1
– pi).
The covariance of outcomes i and j is
–npipj for
distinct i and j.
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Example
The following uses mnpdf to
produce a visualization of a trinomial distribution:
% Compute the distribution
p = [1/2 1/3 1/6]; % Outcome probabilities
n = 10; % Sample size
x1 = 0:n;
x2 = 0:n;
[X1,X2] = meshgrid(x1,x2);
X3 = n-(X1+X2);
Y = mnpdf([X1(:),X2(:),X3(:)],repmat(p,(n+1)^2,1));
% Plot the distribution
Y = reshape(Y,n+1,n+1);
bar3(Y)
set(gca,'XTickLabel',0:n)
set(gca,'YTickLabel',0:n)
xlabel('x_1')
ylabel('x_2')
zlabel('Probability Mass')
Note that the visualization does not show x3,
which is determined by the constraint x1 + x2 + x3 = n.
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