题目

Problem

The random variable X is defined as

X=4A-3B

where A and B are independent and

A \sim N(15,5^2) \qquad B \sim N(10,4^2)

(a) Find $P(X<40)$

The random variable C is such that $C \sim N(20,\sigma^2)$ The random variables $C_1, C_2$ and $C_3$ are independent and each has the same distribution as C The random variable D is defined as

D=\sum_{i=1}^3 C_i

Given that $P(A+B+D<76)=0.2420$ and that A, B and D are independent,

(b) showing your working clearly, find the standard deviation of C

(4)

(6)

(Total for Question 7 is 10 marks)

题目中文翻译

随机变量 X 定义为

X=4A-3B

其中 A 和 B 相互独立，并且

A \sim N(15,5^2) \qquad B \sim N(10,4^2)

(a) 求 $P(X<40)$ 。

随机变量 C 满足 $C \sim N(20,\sigma^2)$ 。随机变量 $C_1, C_2, C_3$ 相互独立，并且都与 C 服从同样的分布。随机变量 D 定义为

D=\sum_{i=1}^3 C_i

已知 $P(A+B+D<76)=0.2420$ ，且 A、B、D 相互独立，

(b) 请清楚写出过程，求 C 的标准差。

IAL 2023 June S3 Q7

题目

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