Convex combination with binomial probabilitiesSuppose that we have some zk&gt;0, k={0,1,⋯,n}. I want to compare weighted averages of zk's when the weights are defined by binomial probabilities.More specifically, for p and q, where p,q∈(0,1), and for some λ∈(0,1), let x=λp+(1−λ)q.In this case, should the following be true for all λ∈(0,1)? max { ∑ k = 0 n ( n k ) p k ( 1 − p ) n − k z k , ∑ k = 0 n ( n k ) q k ( 1 − q ) n − k z k } ≥ ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k z k .I could see that it's true for n = 2 but I can't show that it still holds for any n &gt; 2.

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Convex combination with binomial probabilitiesSuppose that we have some zk&amp;gt;0, k={0,1,⋯,n}. I want to compare weighted averages of zk&#039;s when the weights are defined by binomial probabilities.More specifically, for p and q, where p,q∈(0,1), and for some λ∈(0,1), let x=λp+(1−λ)q.In this case, should the following be true for all λ∈(0,1)?  max            {            ∑          k      =      0              n                                    (                            n        k                              )                          p    k    (  1  −  p      )          n      −      k            z    k    ,      ∑          k      =      0              n                                    (                            n        k                              )                          q    k    (  1  −  q      )          n      −      k            z    k              }        ≥      ∑          k      =      0              n                                    (                            n        k                              )                          x    k    (  1  −  x      )          n      −      k            z    k    .I could see that it&#039;s true for   n  =  2 but I can&#039;t show that it still holds for any   n  &amp;gt;  2.

Maleah Lester · Accepted Answer

Step 1No, this is not the case, not even for   n  =  2. You excluded 0 and 1, but by continuity we can still use them for a simple counterexample:Step 2For   p  =  0,   q  =  1, the left-hand side is zero whereas the right-hand side is positive for any   λ  ∈  (  0  ,  1  ).

Convex combination with binomial probabilities. Suppose that we have some z_k>0, k={0,1,⋯,n}. I want to compare weighted averages of z_k's when the weights are defined by binomial probabilities.

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