function pdf = beta_binomial_pdf ( x, a, b, c ) %*****************************************************************************80 % %% BETA_BINOMIAL_PDF evaluates the Beta Binomial PDF. % % Discussion: % % PDF(X)(A,B,C) = Beta(A+X,B+C-X) % / ( (C+1) * Beta(X+1,C-X+1) * Beta(A,B) ) for 0 <= X <= C. % % This PDF can be reformulated as: % % The beta binomial probability density function for X successes % out of N trials is % % PDF2(X)( N, MU, THETA ) = % C(N,X) * Product ( 0 <= R <= X - 1 ) ( MU + R * THETA ) % * Product ( 0 <= R <= N - X - 1 ) ( 1 - MU + R * THETA ) % / Product ( 0 <= R <= N - 1 ) ( 1 + R * THETA ) % % where % % C(N,X) is the combinatorial coefficient; % MU is the expectation of the underlying Beta distribution; % THETA is a shape parameter. % % A THETA value of 0 ( or A+B --> Infinity ) results in the binomial % distribution: % % PDF2(X) ( N, MU, 0 ) = C(N,X) * MU^X * ( 1 - MU )^(N-X) % % Given A, B, C for PDF, then the equivalent PDF2 has: % % N = C % MU = A / ( A + B ) % THETA = 1 / ( A + B ) % % Given N, MU, THETA for PDF2, the equivalent PDF has: % % A = MU / THETA % B = ( 1 - MU ) / THETA % C = N % % BETA_BINOMIAL_PDF(X)(1,1,C) = UNIFORM_DISCRETE_PDF(X)(0,C-1) % % Licensing: % % This code is distributed under the GNU LGPL license. % % Modified: % % 09 October 2004 % % Author: % % John Burkardt % % Parameters: % % Input, integer X, the argument of the PDF. % % Input, real A, B, parameters of the PDF. % 0.0 < A, % 0.0 < B. % % Input, integer C, a parameter of the PDF. % 0 <= C. % % Output, real PDF, the value of the PDF. % if ( x < 0 ) pdf = 0.0; elseif ( x <= c ) pdf = r8_beta ( a + x, b + c - x ) ... / ( ( c + 1 ) * r8_beta ( x + 1, c - x + 1 ) * r8_beta ( a, b ) ); elseif ( c < x ) pdf = 0.0; end return end