题目

Problem

A researcher repeats, independently, 100 trials of a task. The number of successes is $X$ and the probability of success on each trial is 0.05

(a) Find $P(X = 2)$

(2)

(b) Use a suitable Poisson approximation to estimate $P(X = 2)$

(2)

(d) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

(1)

The random variable $Y$ has a Poisson distribution with mean $\lambda$ such that

$P(Y = 6) = 0.1601 \quad \text{and} \quad P(Y = 7) = 0.1418$

(e) By forming and solving 2 equations, find the value of $\lambda$ to 2 significant figures. Show your working clearly.

(4)

(Total for Question 1 is 11 marks)

题目中文翻译

一位研究者独立重复了 100 次任务试验。成功次数为 $X$ ，每次试验成功的概率为 0.05。

(a) 求 $P(X = 2)$ 。

(b) 使用合适的泊松近似估计 $P(X = 2)$ 。

(d) 说明可以使用泊松分布作为二项分布近似的条件。

随机变量 $Y$ 服从均值为 $\lambda$ 的泊松分布，使得

$P(Y = 6) = 0.1601 \quad \text{且} \quad P(Y = 7) = 0.1418$

(e) 通过建立并求解 2 个方程，求 $\lambda$ 的值（保留 2 位有效数字）。清楚地展示你的计算过程。

（第 1 题共 11 分）

IAL 2025 Jan S2 Q1