Let $F\left( n \right) = \sum\limits_{k = 0}^n {{{\left( {C_n^k{p^k}{{\left( {1 - p} \right)}^{n - k}}} \right)}^2}} $, where $0 \le p \le 1$, prove $F\left( n \right) \ge F\left( {n + 1} \right)$.
How to prove the sum of squared binomial probabilities does not increase when the sample size n increases
Jack
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