2022 Putnam B5

For $0 \leq p \leq 1/2$, let $X_1, X_2, \dots$ be independent random variables such that

$$

X_i = \begin{cases} 1 & \text{with probability $p$,} \\

-1 & \text{with probability $p$,} \\

0 & \text{with probability $1-2p$,}

\end{cases}

$$

for all $i \geq 1$. Given a positive integer $n$ and integers $b, a_1, \dots, a_n$, let $P(b, a_1, \dots, a_n)$ denote the probability that $a_1 X_1 + \cdots + a_n X_n = b$. For which values of $p$ is it the case that

$$

P(0, a_1, \dots, a_n) \geq P(b, a_1, \dots, a_n)

$$

for all positive integers $n$ and all integers $b, a_1, \dots, a_n$?

对于 $0 \leq p \leq 1/2$，令 $X_1, X_2, \dots$ 为独立随机变量，使得

$$

X_i = \begin{cases} 1 & \text{概率为 $p$,} \\

-1 & \text{概率为 $p$,} \\

0 & \text{概率为 $1-2p$,}

\结束{案例}

$$

对于所有 $i \geq 1$。给定正整数 $n$ 和整数 $b, a_1, \dots, a_n$，令 $P(b, a_1, \dots, a_n)$ 表示 $a_1 X_1 + \cdots + a_n X_n = b$ 的概率。 $p$ 的哪些值是这样的

$$

P(0, a_1, \dots, a_n) \geq P(b, a_1, \dots, a_n)

$$

对于所有正整数 $n$ 和所有整数 $b, a_1, \dots, a_n$?