题目

Problem

It is known that $1$ in $5000$ people in Atalia have a certain condition. A random sample of $12\,500$ people from Atalia is chosen for a medical trial. The number having the condition is denoted by $X$ .

(a) Use an appropriate approximating distribution to find $\mathrm{P}(X\leq3)$ .

(3)

(b) Find the values of $\mathrm{E}(X)$ and $\mathrm{Var}(X)$ , and explain how your answers suggest that the approximating distribution used in (a) is likely to be appropriate.

(2)

题目中文翻译

已知 Atalia 每 $5000$ 人中有 $1$ 人患有某种疾病。从 Atalia 随机抽取 $12\,500$ 人参加一项医学试验。患有这种疾病的人数记为 $X$ 。

(a) 用适当的近似分布求 $\mathrm{P}(X\leq3)$ 。

(3)

(b) 求 $\mathrm{E}(X)$ 和 $\mathrm{Var}(X)$ 的值，并说明这些结果为何表明 (a) 中采用的近似分布可能是合适的。

(2)

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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样本量大、患病率低，采用参数2.5的泊松近似并累加0至3。

答题过程

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The binomial model is $X\sim\mathrm{B}(12500,\frac1{5000})$ . Since the sample size is large and the probability is small, use

X\approx\mathrm{Po}\bigg(12500\frac1{5000}\bigg)=\mathrm{Po}(2.5).

Then

\begin{align*} \mathrm{P}(X\leq3)\approx&\,e^{-2.5}(1+2.5)\\[4mm] &\,+e^{-2.5}\bigg(\frac{2.5^2}{2!}+\frac{2.5^3}{3!}\bigg)\\[4mm] =&\,0.757576\ldots\\[4mm] =&\,\boxed{0.758}\quad\text{(3 s.f.).} \end{align*}

(b)

解法一

思路

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本问求原始二项模型的均值与方差，不能把近似泊松的方差2.5当成精确原值。

答题过程

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For the binomial distribution,

\boxed{\mathrm{E}(X)=12500\frac1{5000}=2.5},

and

\begin{align*} \mathrm{Var}(X)=&\,12500\frac1{5000}\bigg(1-\frac1{5000}\bigg)\\[4mm] =&\,\boxed{2.4995}. \end{align*}

The mean and variance are almost equal, which is consistent with a Poisson approximation.

CIE 9709 2023 June Paper 63 Q6

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程