Certainly! Here's the transcription suitable for an educational website:---**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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Answered: Let X1 and X2 be independent and…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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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Use the Suppose in a local Kindergarten through 12th grade (K -12) school district, 42% of the population favor a charter school for grades K through 5. A simple random sample of 144 is surveyed. nd the mean and the standard deviation of X of B(144, 0.42). Round off to 4 decimal places. b. Now approximate X of B(144, 0.42) using the normal approximation with the random variable Y and the table. Round off to 4 decimal places. Y N( C Find the probability that at most 72 favor a charter school using the normal approximation and the table. Round off to z-values up to 2 decimal places.) P(X 68) P(Y > ~ (Z >) e. Find the probability that exactly 63 favor a charter school using the normal approximation and the table. (Round off to z-values up to 2 decimal places.) P(X = 63) P(
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Let x¡, denote a series of n numbers {x;: i=1,2,..,n}: Assume that x; is drawn from a distribution that is N(ux, ož). Show that the sample mean x = E, xị has a variance of ož/n carefully stating any required assumptions i. 1 at each step. ii. Show that the sample mean is an unbiased estimator of ux.
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Find the first two moments about the mean for the random variable X defined by X = -2 with prob. 1/3 3 with prob. 1/2 = 1 with prob. 1/6. %3D
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#6
Let the random variables X and Y have the joint p.m.f. x+ y f (x, y) = 32 x = 1, 2, y = 1,2,3,4. Find the means lx and uy, the variances oy and oý, and the correlation coefficient p. Are X and Y independent or dependent? 6.
Show full answers and steps to this exercise Please don’t use R or excel formula to calculate E(X) and variance. Use normal formula to get to the numbers
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Q1.2 Mean of X A medical practice collects data on its patients with respect to several personal habits. The following table gives the distribution of the number of alcoholic drinks consumed per day in that population of patients. X = # of drinks per day Probability Save Answer Q1.3 Variance of X X = # of drinks per day Probability 0 0.38 Find the mean for this random variable, that is find ux. Show your work in the space below. Save Answer A medical practice collects data on its patients with respect to several personal habits. The following table gives the distribution of the number of alcoholic drinks consumed per day in that population of patients. 0 0.38 Q1.4 Standard Deviation of X X = # of drinks per day Probability 1 0.32 0 0.38 1 0.32 2 0.25 Find the variance for this random variable, that is find (ox)². Only show the first and last term of the work required. 1 2 0.25 A medical practice collects data on its patients with respect to several personal habits. The following table…

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