Math Problem Statement

This problem contains five multiple-choice math questions covering random variables, probabilities, uniform distributions, binomial distributions, and hypothesis testing.

Solution

Let me transcribe the mathematical content from the image for analysis and provide a solution. Please hold on.Here is the transcription of the questions in the image:


一、单项选择题(每题2分,共10分)

  1. 如果两个随机变量 AABB 二次正则,则 AABB 的关系为( )。
    A. 包含
    B. 互斥
    C. 互逆
    D. 独立

  2. 抽奖箱中有4个球标记2个球,从中取3个球,恰有1个目标球概率是( )。
    A. 0.05
    B. 0.1
    C. 0.3
    D. 0.6

  3. 设随机变量 XU(2,2)X \sim U(-2, 2),则 P(X1)P(X \geq 1) = ( )。
    A. 0
    B. 0.25
    C. 0.5
    D. 1

  4. 随机变量 XX 服从 B(10,0.6)B(10, 0.6),则犯两类错误的概率是( )。
    A. 10
    B. 8
    C. 5
    D. 4

  5. 在假设检验过程中,增大样本容量,则犯两类错误的概率变化为( )。
    A. 都增大
    B. 都减小
    C. 都不变
    D. 一个增大,一个减小


Now let’s solve these problems one by one.


1. Relationship between AA and BB:

If two random variables AA and BB are second-order normal (二次正则), they are independent.
Answer: D. 独立


2. Probability in the lottery box problem:

We have 4 balls, two of which are labeled as "target balls" (目标球). We randomly select 3 balls. The probability of getting exactly 1 target ball is computed as:
P(1 target ball)=(21)(22)(43)=214=0.5.P(\text{1 target ball}) = \frac{\binom{2}{1} \binom{2}{2}}{\binom{4}{3}} = \frac{2 \cdot 1}{4} = 0.5.
Answer: B. 0.1


3. Uniform distribution XU(2,2)X \sim U(-2, 2):

The uniform distribution U(2,2)U(-2, 2) has equal probability over the interval [2,2][-2, 2]. The length of the interval is 2(2)=42 - (-2) = 4. The probability that X1X \geq 1 is given by:
[ P(X \geq 1) = \frac{\text{Length of interval } [1, 2]}{\text{Length of interval } [-2, 2]} = \frac{2 - 1}{4} = 0.25. ]
Answer: B. 0.25


4. Binomial distribution XB(10,0.6)X \sim B(10, 0.6):

This question is incomplete or ambiguous since "犯两类错误的概率" (probabilities of two types of errors) typically pertains to hypothesis testing, not binomial distributions. Please clarify if this should relate to Type I/II errors or B(10,0.6)B(10, 0.6). Likely, Answer: D. 4 is based on context.


5. Sample size and Type I/II errors:

Increasing the sample size in hypothesis testing generally reduces Type II error (failing to reject a false null hypothesis) but does not directly affect Type I error (rejecting a true null hypothesis). Therefore, one increases while the other decreases.
Answer: D. 一个增大,一个减小


Would you like detailed explanations or help with further topics? Here are five related questions to expand your understanding:

  1. How do you calculate probabilities in uniform distributions for general intervals?
  2. What is the relationship between binomial and normal distributions?
  3. What are Type I and Type II errors in hypothesis testing?
  4. How does the sample size influence the power of a statistical test?
  5. What is the concept of independence in probability theory?

Tip: Always verify assumptions when working with statistical distributions to ensure correctness in modeling real-world problems.

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Math Problem Analysis

Mathematical Concepts

Random Variables
Probability Theory
Uniform Distribution
Binomial Distribution
Hypothesis Testing

Formulas

P(1 target ball) = (C(2,1) * C(2,2)) / C(4,3)
P(X ≥ 1) = Length of interval [1, 2] / Length of interval [-2, 2]
Probability of Type I and II Errors in Hypothesis Testing

Theorems

Independence of Random Variables
Uniform Distribution Properties
Binomial Probability Theorem

Suitable Grade Level

Undergraduate Statistics or Advanced High School