Let X1 and X2 be independent and identical normal random variables with common mean 3 and standard deviation 1. Consider the sample mean U = Xi+X2 a. What is the moment-generating function my (t) of U? [Hint: Refer to Proposition 3.1, mu (t) = mx,(G)mx, (t) - ] b. What is the distribution of U? c. Find the probability that the observed sample mean is between 2.8 and 3.2.
Let X1 and X2 be independent and identical normal random variables with common mean 3 and standard deviation 1. Consider the sample mean U = Xi+X2 a. What is the moment-generating function my (t) of U? [Hint: Refer to Proposition 3.1, mu (t) = mx,(G)mx, (t) - ] b. What is the distribution of U? c. Find the probability that the observed sample mean is between 2.8 and 3.2.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Independent Normal Random Variables Analysis**
*Consideration and Analysis of Independent Normal Random Variables and Their Sample Mean*
Let \( X_1 \) and \( X_2 \) be independent and identical normal random variables with a common mean of 3 and a standard deviation of 1. We are examining the sample mean \( U = \frac{X_1 + X_2}{2} \).
**a. Moment-Generating Function of \( U \):**
What is the moment-generating function \( m_U(t) \) of \( U \)?
*Hint*: Refer to Proposition 3.1 for guidance.
\[ m_U(t) = m_{X_1}\left(\frac{t}{n}\right)m_{X_2}\left(\frac{t}{n}\right) \]
**b. Distribution of \( U \):**
What is the distribution of the sample mean \( U \)?
**c. Probability Calculations:**
Find the probability that the observed sample mean is between 2.8 and 3.2.
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*Note: The analysis requires understanding the properties of moment-generating functions and the distributions of normal random variables.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9ad9a17-86a2-4d23-8aeb-c537a78d9db8%2Fa7880149-ef95-4ef3-a747-47a7bd754d5f%2Ftrht8hr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here's the transcription suitable for an educational website:
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**Independent Normal Random Variables Analysis**
*Consideration and Analysis of Independent Normal Random Variables and Their Sample Mean*
Let \( X_1 \) and \( X_2 \) be independent and identical normal random variables with a common mean of 3 and a standard deviation of 1. We are examining the sample mean \( U = \frac{X_1 + X_2}{2} \).
**a. Moment-Generating Function of \( U \):**
What is the moment-generating function \( m_U(t) \) of \( U \)?
*Hint*: Refer to Proposition 3.1 for guidance.
\[ m_U(t) = m_{X_1}\left(\frac{t}{n}\right)m_{X_2}\left(\frac{t}{n}\right) \]
**b. Distribution of \( U \):**
What is the distribution of the sample mean \( U \)?
**c. Probability Calculations:**
Find the probability that the observed sample mean is between 2.8 and 3.2.
---
*Note: The analysis requires understanding the properties of moment-generating functions and the distributions of normal random variables.*
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