题目

Problem

The random variable $X \sim B(10, \, p)$

(a) (i) Write down an expression for $P(X = 3)$ in terms of $p$

(ii) Find the value of $p$ such that $P(X = 3)$ is 16 times the value of $P(X = 7)$

(4)

The random variable $Y \sim \text{Po}(\lambda)$

(b) Find the value of $\lambda$ such that $P(Y = 3)$ is 5 times the value of $P(Y = 5)$

(3)

The random variable $W \sim B(n, \, 0.4)$

(c) Find the value of $n$ and the value of $\alpha$ such that $W$ can be approximated by the normal distribution, $N(32, \, \alpha^2)$

(3)

(Total for Question 2 is 10 marks)

题目中文翻译

随机变量 $X \sim B(10, \, p)$

(a) (i) 写出 $P(X = 3)$ 关于 $p$ 的表达式

(ii) 求 $p$ 的值，使得 $P(X = 3)$ 是 $P(X = 7)$ 的 16 倍

随机变量 $Y \sim \text{Po}(\lambda)$

(b) 求 $\lambda$ 的值，使得 $P(Y = 3)$ 是 $P(Y = 5)$ 的 5 倍

随机变量 $W \sim B(n, \, 0.4)$

（第 2 题共 10 分）

IAL 2026 Jan S2 A Q2