You are playing a board game in which severity of a plenty is determined by rolling three dice and adding spots on the up-face. The dice are all balanced so that each face is equally likely, and the three dice fall independently.

Question

You are playing a board game in which severity of a plenty is determined by rolling three dice and adding spots on the up-face. The dice are all balanced so that each face is equally likely, and the three dice fall independently.

If X1, X2, and X3 are the number of spots on the up-face of the three dice, then X= X1+X2+X3.
Use this fact to find the mean μx and the standard deviation σx without finding the distribution of of X.
(Start with the distribution of each of the Xi.)

Please help me!!!! I have been working one this for over and hour and its due tomorrow.

THANKS in ADVANCE!!!!

Answer 1

See my post to MAI on tuesday, December 16.

Answer 2

To find the mean (average) and standard deviation of X, we can start by understanding the distribution of each of the Xi, the number of spots on the up-face of each die.

Since each die is balanced and equally likely to land on any face, the distribution of each Xi follows a uniform distribution. A uniform distribution means that each value in the range of possible outcomes is equally likely.

In this case, each Xi can take values from 1 to 6, with equal probabilities for each value.

Now, to find the mean of X, denoted by μx, we need to find the expected value, which is the average of X.

The mean of each Xi, denoted by μi, can be calculated by taking the sum of all possible outcomes divided by the total number of outcomes. For a uniform distribution, this is simply the sum of all values divided by the total number of values:

μi = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5

Since X is the sum of the three dice, the mean of X is the sum of the means of each Xi:

μx = μ1 + μ2 + μ3 = 3.5 + 3.5 + 3.5 = 10.5

Therefore, the mean of X is 10.5.

Next, let's move on to finding the standard deviation of X, denoted by σx.

The variance of each Xi, denoted by σi^2, is a measure of how much the values in the distribution deviate from the mean. For a uniform distribution, the variance can be calculated by the following formula:

σi^2 = ((1 - μi)^2 + (2 - μi)^2 + ... + (6 - μi)^2) / 6

Using this formula, we find that the variance of each Xi is:

σi^2 = ((1 - 3.5)^2 + (2 - 3.5)^2 + ... + (6 - 3.5)^2) / 6 = 2.9167

Since X is the sum of independent random variables, the variance of X is the sum of the variances of each Xi:

σx^2 = σ1^2 + σ2^2 + σ3^2 = 2.9167 + 2.9167 + 2.9167 = 8.7501

Finally, the standard deviation of X, σx, is the square root of the variance:

σx = √(8.7501) ≈ 2.958

Therefore, the mean of X is approximately 10.5, and the standard deviation of X is approximately 2.958.

Remember, it is important to understand the concepts and formulas involved, as this will help you solve similar problems in the future. Good luck!