题目

Problem

Two fair 6-sided dice with faces labelled 1, 2, 3, 4, 5, 6 are thrown. The two scores are noted. The random variable $X$ is defined as follows.

If the two scores are equal, $X=0$
If the scores are not equal, $X$ is the larger score minus the smaller score

(a) Draw up the probability distribution table for $X$ .

(3)

(b) Find $\mathrm{E}(X)$ and $\operatorname{Var}(X)$ .

(3)

题目中文翻译

掷两枚均匀的六面骰子，每枚骰子的各面分别标有 1、2、3、4、5、6。记下两个点数。随机变量 $X$ 定义如下。

如果两个点数相等，则 $X=0$
如果两个点数不相等，则 $X$ 为较大的点数减去较小的点数

(a) 列出 $X$ 的概率分布表。

(3)

(b) 求 $\mathrm{E}(X)$ 和 $\operatorname{Var}(X)$ 。

(3)

原卷通用作答要求

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

解答

(a)

解法一

思路

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两骰有36个等可能有序结果。差为0有6个；正差为x时，较小数可从1选到6减x，且两骰次序可交换。

答题过程

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There are 36 equally likely ordered outcomes. For $X=0$ , the scores must be equal, giving six outcomes.

For $x=1,2,3,4,5$ , choose the smaller score in $6-x$ ways and choose which die shows it in two ways. Thus $\mathrm{P}(X=x)=\frac{2(6-x)}{36}$ .

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$\mathrm{P}(X=x)$	$\frac6{36}$	$\frac{10}{36}$	$\frac8{36}$	$\frac6{36}$	$\frac4{36}$	$\frac2{36}$

(b)

解法一

思路

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用分布表求一阶矩及二阶矩，最后把均值与方差明确区分。

答题过程

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\begin{align*} \mathrm{E}(X)=&\,\frac{10+16+18+16+10}{36} =\boxed{\frac{35}{18}},\\[4mm] \mathrm{E}(X^2)=&\,\frac{10+32+54+64+50}{36} =\frac{35}{6}. \end{align*}

Therefore,

\begin{align*} \operatorname{Var}(X) =&\,\frac{35}{6}-\biggl(\frac{35}{18}\biggr)^2\\[4mm] =&\,\frac{1890-1225}{324}\\[4mm] =&\,\boxed{\frac{665}{324}}. \end{align*}

CIE 9709 2025 June Paper 55 Q1

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程