随机变量 $Y$ 的概率母函数 $G_Y(t)$ 为

(a) 求 $\operatorname{Var}(Y)$。

随机变量 $Y$ 是随机变量 $X$ 的两个独立观测值之和。

(b) 求 $X$ 的概率母函数，并将答案表示为 $t$ 的多项式。

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。

从生成函数在 $1$ 处的两阶导数求均值和阶乘矩。注意方差公式还要加回均值。

Differentiating gives

Thus $G_Y'(1)=2.2$ and $G_Y''(1)=3.62$. Using $\operatorname{Var}(Y)=G_Y''(1)+G_Y'(1)-\big(G_Y'(1)\big)^2$, we obtain $\operatorname{Var}(Y)=3.62+2.2-2.2^2=0.98$.

独立的两个相同变量使 $G_Y$ 成为完全平方。用最低、最高次项的系数确定 $G_X$ 的两端，再找中间项。

Since $G_Y=G_X^2$, the degree of $G_X$ is $2$. Write $G_X(t)=a+bt+ct^2$, with nonnegative coefficients. Comparing the constant and quartic terms gives $a^2=0.04$ and $c^2=0.09$, so $a=0.2$ and $c=0.3$. The linear term gives $2ab=0.2$, hence $b=0.5$. Therefore

Squaring it reproduces all coefficients of $G_Y$.

将四次多项式识别为两个一次因子的平方。两个同分布独立变量相加使母函数平方，最后展开成题目要求的多项式。

The given polynomial factorises as

Since $G_Y(t)=G_X(t)^2$ and a probability generating function has nonnegative coefficients,

The coefficients sum to $1$, as required.