Probability - Sum of Squares of Rolls of a Die

In summary, the problem involves rolling a fair die n times and calculating the sum of squares of the rolls, denoted as Sn. The mean and variance of sqrt(n) * (Sn/n - u) are being sought, where u is the mean of Yn/n. The solution involves looking into the law of large numbers and possibly using the definition of expected value and variance. However, it is not specified what Yn represents.
  • #1
jchiz24
3
0

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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  • #2
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]?
 

Related to Probability - Sum of Squares of Rolls of a Die

1. What is the probability of getting a sum of 10 when rolling two dice?

The probability of getting a sum of 10 when rolling two dice is 1/12 or approximately 8.33%. This can be calculated by finding the number of possible outcomes that result in a sum of 10 (3 and 4, 4 and 3, 5 and 2, and 2 and 5) out of the total number of possible outcomes (36). Therefore, 4/36 = 1/9.

2. Is the probability of getting a sum of 7 the same as getting a sum of 8 when rolling two dice?

No, the probabilities are not the same. The probability of getting a sum of 7 when rolling two dice is 1/6 or approximately 16.67%, while the probability of getting a sum of 8 is 5/36 or approximately 13.89%. This is because there are more possible combinations that result in a sum of 7 (6) compared to a sum of 8 (5).

3. How does the sum of squares of rolls of a die relate to probability?

The sum of squares of rolls of a die is a method used to calculate the probability of a certain outcome in a dice game. It is based on the assumption that all outcomes are equally likely, and the probability of each outcome can be calculated by dividing the number of ways to obtain that outcome by the total number of possible outcomes.

4. What is the formula for calculating the sum of squares of rolls of a die?

The formula for calculating the sum of squares of rolls of a die is (n(n+1)(2n+1))/6, where n is the number of dice being rolled. For example, if rolling two dice, the formula would be (2(2+1)(2(2)+1))/6 = 5/6 = 0.8333.

5. Can the sum of squares of rolls of a die be used to determine the likelihood of winning a dice game?

Yes, the sum of squares of rolls of a die can be used to determine the likelihood of winning a dice game. By calculating the probability of different outcomes and comparing them to the rules of the game, one can determine the likelihood of winning and make informed decisions during the game.

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