Exponential-logarithmic distribution: Difference between revisions

Revision as of 04:21, 6 September 2009

Revision as of 04:24, 6 September 2009

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The moment generating function of <math>X</math> is determined from pdf by

direct integration and is given by

: <math>M_X(t) = E(e^{tX}) = -\frac{\beta(1-p)}{\ln p (\beta-t)} hypergeom_\operatorname{hypergeom}_{2,1}\left(\left[1,\frac{\beta-t}{\beta}\right],\left[\frac{2\beta-t}{\beta}\right],1-p\right),</math><br>

where hypergeom<mathsub>hypergeom(.)2,1</mathsub> is hypergeometric function. This function

is also known as ''Barnes's extended hypergeometric function''. The

definition of <math>F_{p,q}({n,d},\lambda)</math> is<br>

: <math>F_{p,q}({n,d},\lambda)=\sum_{k=0}^\infty \frac{\lambda^k \prod_{i=1}^p\Gamma(n_i+k)\Gamma^{-1}(n_i)}{\Gamma(k+1)\prod_{i=1}^q\Gamma(d_i+k)\Gamma^{-1}(d_i)}</math>

<br>where <math>{n}=[n_1, n_2, ..., n_p]</math>, <math>p</math> is the number of

operands of <math>{n}</math>, <math>{d}=[d_1, d_2, ..., d_q]</math> and <math>q</math> is

<br>where <math>{n}=[n_1, n_2, ..., n_p]</math>, <math>p</math> is the number of

operands of <math>{n}</math>, <math>{d}=[d_1, d_2, ...\dots, d_q]</math> and <math>q</math> is

the number of operands of <math>{d}</math>. Generalized hypergeometric

function is quickly evaluated and readily available in standard