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.

puntdald8 2022-09-13 Answered
Convex combination with binomial probabilities
Suppose that we have some zk>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 > 2.
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Answers (1)

Maleah Lester
Answered 2022-09-14 Author has 12 answers
Step 1
No, 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 2
For p = 0, q = 1, the left-hand side is zero whereas the right-hand side is positive for any λ ( 0 , 1 ).

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