Is the probability model given by the following distribution function F p ( x ) = { ∑ k = 1 ⌊ x ⌋ p ( 1 − p ) k − 1 if x ≥ 1 0 else dominated? (in which case exhibit a dominant measure and give the Radon-Nikodym derivative).I know the definition of a dominated model:A model ( P θ ) θ is dominated if there exists a measure μ &gt; 0 and σ −−finite such that P θ is dominated by μ and P θ is absolutely continuous w.r.t μBut I don't know how to show that the model given by this distribution function is dominated.

Question

Is the probability model given by the following distribution function      F          p        (  x  )  =      {                                        ∑                          k              =              1                                      ⌊              x              ⌋                                p          (          1          −          p                      )                          k              −              1                                                 if           x          ≥          1                                      0                           else                         dominated? (in which case exhibit a dominant measure and give the Radon-Nikodym derivative).I know the definition of a dominated model:A model   (      P          θ            )          θ       is dominated if there exists a measure   μ  &amp;gt;  0 and   σ  −−finite such that       P          θ       is dominated by   μ and       P          θ       is absolutely continuous w.r.t   μBut I don&#039;t know how to show that the model given by this distribution function is dominated.

Makai Blackwell · Accepted Answer

The measure you are looking for is the counting measure on the integers (or on the support of the measure associated to   F). Notice that since       F    p   takes all its mass on             Z              ≥      1      , if the counting measure of a set is zero, then so is its probability under the measure associated to       F    p  .

Is the probability model given by the following distribution function F <mrow class=

Answered question

Answer & Explanation

New Questions in Pre-Algebra