随机变量 $X$ 只在 $0$ 与 $6$ 之间取值，其概率密度函数 $f(x)$ 如图所示，其中 $a$ 是常数。

已知 $\mathrm{E}(X)=2$，求 $\operatorname{Var}(X)$。

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。

图中密度下方面积是三角形，总面积为 $1$；确定高度后写出直线方程，再求二阶矩。

The area under the density is a triangle, so

The line through $(0,1/3)$ and $(6,0)$ has gradient $-1/18$. Thus

This applies for $0\le x\le6$.

The second moment is

Using the given $\mathrm E(X)=2$,

也可先保留密度高度 $a$，利用已给的期望反求 $a$，再求二阶矩；最后检查密度总积分为 $1$。

The density line through $(0,a)$ and $(6,0)$ is

This applies for $0\le x\le6$.

Using the given mean,

Hence $a=1/3$. This also gives total probability $\int_0^6 f(x)\,\mathrm{d}x=3a=1$, as required. Now

Therefore $\boxed{\operatorname{Var}(X)=6-2^2=2}$.