Math Problem Statement
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)
-
如果两个随机变量 与 二次正则,则 与 的关系为( )。
A. 包含
B. 互斥
C. 互逆
D. 独立 -
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 -
设随机变量 ,则 = ( )。
A. 0
B. 0.25
C. 0.5
D. 1 -
随机变量 服从 ,则犯两类错误的概率是( )。
A. 10
B. 8
C. 5
D. 4 -
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 and :
If two random variables and 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:
Answer: B. 0.1
3. Uniform distribution :
The uniform distribution has equal probability over the interval . The length of the interval is . The probability that 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 :
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 . 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:
- How do you calculate probabilities in uniform distributions for general intervals?
- What is the relationship between binomial and normal distributions?
- What are Type I and Type II errors in hypothesis testing?
- How does the sample size influence the power of a statistical test?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Probability and Statistics Problems with Binomial and Geometric Distributions
Comprehensive Solutions for Probability, Distributions, and Markov Chain Questions
Solve Probability and Distribution Problems for Discrete and Continuous Variables
Probability and Statistics Problems: Binomial and Normal Distributions
Understanding Probability and Variable Types: Analyzing X