掷骰期望比较：乘积、平方、五次中位数平方

Die Rank

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

独立掷 5 次公平六面骰得 $X_1,\dots,X_5$ 。比较下列三者从大到小的顺序：

a) $\mathbb{E}[X_1X_2]$ ；

b) $\mathbb{E}[X_1^2]$ ；

c) $\mathbb{E}[X_{(3)}^2]$ ，其中 $X_{(3)}$ 为 5 次结果的中位数。

用 a=1,b=2,c=3 的数字拼接输出顺序（例如 c>b>a 输出 321）。

Let $X_1, \dots, X_5$ be the outcomes of 5 independent rolls of a fair 6-sided die with values $1-6$ on each side. Rank each of the following from largest to smallest in terms of value: a) $\mathbb{E}[X_1 X_2]$ ; b) $\mathbb{E}[X_1^2]$ ; c) $\mathbb{E}[X_{(3)}^2]$ , where $X_{(3)}$ is the median of the 5 die rolls.

Let $a = 1, b = 2,$ and $c = 3$ . Answer with the integer corresponding to the concatenation of the order from largest to smallest. For example, if you believe $c > b > a$ , answer with 321.

解析

可算得：

\mathbb{E}[X_1X_2]=\mathbb{E}[X_1]\mathbb{E}[X_2]=3.5^2=12.25,

\mathbb{E}[X_1^2]=\frac{1^2+2^2+3^2+4^2+5^2+6^2}{6}\approx 15.17.

而中位数更集中在中间偏大的值，其平方期望介于两者之间，因此顺序为 $b>c>a$ ，对应 231。

Original Explanation

We can calculate $\mathbb{E}[X_1 X_2] = \mathbb{E}[X_1] \mathbb{E}[X_2] = (3.5)^2 = 12.25$ by direct calculation via independence of the rolls. Furthermore, we can calculate $\mathbb{E}[X_1^2] = \frac{1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2}{6} \approx 15.17$ , so we know that $b > a$ . We now need to find where $c$ lies relative to these. It's worth just thinking about this intuitively, as the calculations are quite difficult.

By inspection, we note that the distribution here is skewed towards the middle (we would need 3 or 6s or 3 or 1s to get either extreme). Note that in the sum for $\mathbb{E}[X_1^2]$ , we gain much more through the squared 6 than we lose in the squared 1. This ranks it below $\mathbb{E}[X_1^2]$ .

To compare to $\mathbb{E}[X_1 X_2]$ , we note if either one of the dice is low, it reduces our product greatly. This reduction occurs with very high probability, whereas the median is more likely to reach at least a moderate value (say 3 or 4), so the median is skewed more right. Therefore, we get that $b > c > a$ , corresponding to the answer 231.