I would like to maximize the Conﬂuent Hypergeometric Distribution in order to apply a Rejection sampling. The formula of the distribution is f ( x ; a , b , c ) = K x a − 1 ( 1 − x ) b − 1 e − c x where 0 ≤ x ≤ 1 and a , b &gt; 0 , c ∈ R and let us assume that K is constant in terms of x. I would like to maximize this distribution in order to find an upper bound C and apply the rejection sampling.

Question

I would like to maximize the Conﬂuent Hypergeometric Distribution in order to apply a Rejection sampling. The formula of the distribution is  f  (  x  ;  a  ,  b  ,  c  )  =  K      x          a      −      1        (  1  −  x      )          b      −      1            e          −      c      x      where   0  ≤  x  ≤  1 and   a  ,  b  &amp;gt;  0  ,  c  ∈      R  and let us assume that K is constant in terms of x. I would like to maximize this distribution in order to find an upper bound C and apply the rejection sampling.

Cordell Crosby · Accepted Answer

d      f      (      x      )              d      x        =      {          x              a        −        2              (    1    −    x          )              b        −        2                    e              −        c        x                    (      −      a      x      +      a      −      x      (      b      +      c      −      2      )      +      c              x        2            −      1      )        }  and set to 0 gives the real solutions:  x  =            ±              (        a        +        b        +        c        −        2                  )          2                +        4        (        1        −        a        )        c            +      a      +      b      +      c      −      2              2      c

il2k3s2u7 · Accepted Answer

To maximize   f  (  x  ;  a  ,  b  ,  c  ) with respect to   x, it suffices to maximize its logarithm. Since   K is constant, that means we need to maximize  (  a  −  1  )  log  ⁡  x  +  (  b  −  1  )  log  ⁡  (  1  −  x  )  −  c  x  .You can find the stationary points by setting the derivative to 0, i.e.            a      −      1        x    −            b      −      1              1      −      x        =  c  .If you multiply this out you obtain a quadratic equation for   x.

I would like to maximize the Conﬂuent Hypergeometric Distribution in order to apply a Rejection samp

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