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.)

Answer 1

assistance needed

Answer 2

What you have here is a "Distribution of Linear Combinations" problem.

Lets start with the expected mean of Xi. There is a 1/6 prob of rolling either a 1,2,..6. So mean of X1 is (1/6)*(1+2+3+4+5+6) = 3.5. Since all the dice are the same, mean X1= mean X2 = mean X3 = 3.5 Since each die is thrown the same number of times, the expected mean for X = (mean X1)+(mean X2) + (mean X3) = 3.5+3.5+3.5 = 10.5

Now for the variance of X1. The expected variace for X1 is ((1-3.5)^2+(2-3.5)^2 + ...(6-3.5)^2 ) divided by 6. I get Var X1 = 2.9167 and SD X1=1.708.

Since X1,X2,X3 are independent and since each die contributes 1/3 towards the total, SD X = sqrt(Var X1 + Var X2 + Var X3) = sqrt(3*2.9169) = 2.958

Answer 3

(I wonder how I got that strange looking title to my first response???)

Answer 4

To find the mean (μx) and standard deviation (σx) of X without explicitly finding the distribution of X, we can use the properties of expected value and variance.

First, we need to find the mean (μi) and variance (σi^2) of each Xi.

Since each die is balanced, each face has an equal probability of appearing. Hence, the expected value of each Xi is given by the sum of all possible outcomes divided by the number of outcomes. In this case, each die has 6 possible outcomes, ranging from 1 to 6. Therefore, the mean of each Xi is:

μi = (1+2+3+4+5+6)/6 = 3.5

Next, we need to find the variance (σi^2) of each Xi. The variance measures the spread or variability of a distribution.
The variance of a single die can be calculated using the formula:

σi^2 = Σ[Xi - μi)^2 * P(Xi)]

where P(Xi) is the probability of Xi occurring.

Since each face has an equal chance of appearing, the probability of each face is 1/6. Therefore, the variance of each Xi is:

σi^2 = Σ[(Xi - 3.5)^2 * (1/6)]

Now we can calculate the variance of X, denoted as σx^2, using the properties of variances. For independent random variables, the variance of the sum of the variables is equal to the sum of their variances. Since we have three independent dice, the variance of X is:

σx^2 = Σ[σi^2] + Σ[σj^2] + Σ[σk^2]

where i, j, and k refer to the three independent dice.

Using the variance of each Xi calculated previously, we have:

σx^2 = σ1^2 + σ2^2 + σ3^2

Finally, we can find the standard deviation (σx) by taking the square root of the variance:

σx = sqrt(σx^2)

So, to find the mean (μx) and standard deviation (σx) of X without explicitly finding the distribution of X, we just need to calculate the mean and variance of each Xi, and then sum them up as shown above.