Deviations from the mean of a sum of independent, non-identically-distributed Bernoulli variables, continued

Continuing the investigation initiated here, and applying the same notation.

Proposition 2:

For every l, l >= ES, P(S ≥ l) is maximized when q₁ = ··· = q_n-mz = q = ES / (n – m_z), and q_{n – mz} + 1 = ··· = q_n = 0, for some m_z.

That is, in the terminology of Proposition 1, for all non-trivial cases, m_o = 0, and thus the probability of deviation is maximized by some Bernoulli variable (rather than a shifted Bernoulli variable).

Advertisement