Probability - Sum of Squares of Rolls of a Die

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SUMMARY

The discussion focuses on calculating the mean and variance of the expression sqrt(n) * (Sn/n - u), where Sn is the sum of squares of n rolls of a fair die and u is the mean of Yn/n. Participants explore the implications of the law of large numbers and consider whether to approach the problem through the definitions of expected value and variance. The notation used includes Xi for individual rolls and Sn for the cumulative sum of squares, emphasizing the statistical analysis of die rolls.

PREREQUISITES
  • Understanding of probability theory and random variables
  • Familiarity with the law of large numbers
  • Knowledge of expected value and variance calculations
  • Basic concepts of statistical distributions
NEXT STEPS
  • Study the law of large numbers in detail
  • Learn about calculating expected value and variance for discrete random variables
  • Explore the properties of the sum of squares in probability distributions
  • Investigate the Central Limit Theorem and its applications in statistics
USEFUL FOR

Students in statistics or probability courses, mathematicians interested in statistical analysis, and anyone studying the behavior of random variables in relation to die rolls.

jchiz24
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Homework Statement


Roll a fair die n times. Let Sn denote the sum of squares of the rolls. Thus, Sn is the sum of Xi^2, where Xi represents one roll.

What are the mean and variance of sqrt(n) * (Sn/n - u), where u is the mean of Yn/n

Homework Equations




The Attempt at a Solution


No real revelation yet, but looking into law of large numbers. Not sure if should pursue problem directly via definition of expected value and variance...
 
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jchiz24 said:

Homework Statement


Roll a fair die n times. Let Sn denote the sum of squares of the rolls. Thus, Sn is the sum of Xi^2, where Xi represents one roll.

What are the mean and variance of sqrt(n) * (Sn/n - u), where u is the mean of Yn/n

Homework Equations




The Attempt at a Solution


No real revelation yet, but looking into law of large numbers. Not sure if should pursue problem directly via definition of expected value and variance...

So you are calling the results of individual rolls [itex]X_i[/itex], and

[tex] S_n = X_1^2 + X_2^2 + \cdots X_n^2[/tex]

You give the definition that [itex]u[/itex] is the mean of

[tex] \frac{Y_n}{n}[/tex]

What is [itex]Y_n[/itex]?
 

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