题目

Long Conversation 1Section BDirections: In this section, you will hear two long conversations. At the end of each conversation, you will hear four questions. Both the conversation and the questions will be spoken only once. After you hear a question, you must choose the best answer from the four choices marked A), B), C) and D). Then mark the corresponding letter on Answer Sheet 1 with a single line through the centre.Questions 8 to 11 are based on the conversation you have just heard.(1) (单选题)8.A.She got a well-paying job in a bank.B.She received a bonus unexpectedly.C.She received her first monthly salary.D.She got a pay raise for her performance.(2) (单选题)9.A.Several years ago.B.Two decades ago.C.Right after graduation.D.Just last month.(3) (单选题)10.A.He sent a small check to his parents.B.He took a few of his friends to a gym.C.He immediately deposited it in a bank.D.He treated his parents to a nice meal.(4) (单选题)11.A.Buy some professional clothes.B.udget her salary carefully.C.Join her colleagues for gym exercise.D.Visit her former university campus.A. 二维离散型随机变量相互独立，等价于其联合分布律在每个取值点都等于边缘分布律的乘积；B. 二维随机变量相互独立，等价于其联合分布函数在平面上每一点都等于边缘分布函数的乘积；C. 若随机变量X1, X2,X3相互独立，则sin(X1)与X2＋X3也相互独立。D. 二维连续型随机变量相互独立，等价于其联合概率密度在平面上每一点都等于边缘概率密度的乘积；

Long Conversation 1Section BDirections: In this section, you will hear two long conversations. At the end of each conversation, you will hear four questions. Both the conversation and the questions will be spoken only once. After you hear a question, you must choose the best answer from the four choices marked A), B), C) and D). Then mark the corresponding letter on Answer Sheet 1 with a single line through the centre.Questions 8 to 11 are based on the conversation you have just heard.(1) (单选题)8.A.She got a well-paying job in a bank.B.She received a bonus unexpectedly.C.She received her first monthly salary.D.She got a pay raise for her performance.(2) (单选题)9.A.Several years ago.B.Two decades ago.C.Right after graduation.D.Just last month.(3) (单选题)10.A.He sent a small check to his parents.B.He took a few of his friends to a gym.C.He immediately deposited it in a bank.D.He treated his parents to a nice meal.(4) (单选题)11.A.Buy some professional clothes.B.udget her salary carefully.C.Join her colleagues for gym exercise.D.Visit her former university campus.
  • A. 二维离散型随机变量相互独立，等价于其联合分布律在每个取值点都等于边缘分布律的乘积；
  • B. 二维随机变量相互独立，等价于其联合分布函数在平面上每一点都等于边缘分布函数的乘积；
  • C. 若随机变量X1, X2,X3相互独立，则sin(X1)与X2＋X3也相互独立。
  • D. 二维连续型随机变量相互独立，等价于其联合概率密度在平面上每一点都等于边缘概率密度的乘积；

题目解答

答案

1、C2、B3、D4、C

解析

本题考查二维随机变量独立性的判定，涉及离散型、连续型随机变量的独立性定义，以及独立性的保持性质。解题核心在于：

  1. 独立性的定义：离散型、连续型的独立性均要求联合分布（律/密度）等于边缘分布（律/密度）的乘积；
  2. 独立性的传递性：若随机变量独立，则其函数变换后仍保持独立性；
  3. 独立性的反例：需注意某些函数变换可能破坏独立性（如非线性变换）。

选项A

二维离散型随机变量独立的定义是：联合分布律在每个取值点等于边缘分布律的乘积。
结论：正确。

选项B

二维随机变量独立的定义是：联合分布函数在平面上每一点等于边缘分布函数的乘积。
结论：正确。

选项C

若 $X_1, X_2, X_3$ 相互独立，则 $\sin(X_1)$ 与 $X_2 + X_3$ 是否独立？
关键点：

  • $X_1$ 与 $X_2 + X_3$ 独立（因 $X_1$ 与 $X_2, X_3$ 均独立）；
  • 但 $\sin(X_1)$ 是非线性变换，可能与 $X_2 + X_3$ 产生依赖关系。
    反例：若 $X_1$ 服从均匀分布，$X_2, X_3$ 服从正态分布，则 $\sin(X_1)$ 的周期性可能与 $X_2 + X_3$ 的分布产生关联，破坏独立性。
    结论：错误。

选项D

二维连续型随机变量独立的定义是：联合概率密度在平面上每一点等于边缘概率密度的乘积。
结论：正确。