91Ó°ÊÓ

To calculate the mean, we will sum up all the possible outcomes and divide by the total number of outcomes (8 in this case). The mean (µ) can be represented as: \[\mu = \frac{1}{n}\sum_{i=1}^n x_i\] Where \(n\) is the total number of outcomes, \(x_i\) are the outcomes, and the sum runs from 1 to 8 for our eight-sided die. \[\mu = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8}{8}\] Calculate the sum of the outcomes: \[\mu = \frac{36}{8} = 4.5\] The mean of this experiment is 4.5.

To calculate the variance, we will first find the squared difference between each outcome and the mean, and then find the average of these squared differences. The variance (σ²) can be represented as: \[\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2\] Using the mean (4.5) and the sum of outcomes from Step 1, we can calculate the variance for this experiment: \[\sigma^2 = \frac{(1-4.5)^2 + (2-4.5)^2 + (3-4.5)^2 + (4-4.5)^2 + (5-4.5)^2 + (6-4.5)^2 + (7-4.5)^2 + (8-4.5)^2}{8}\] Calculate the squared differences: \[\sigma^2 = \frac{12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25}{8}\] Add up all the squared differences and divide by the total number of outcomes: \[\sigma^2 = \frac{42}{8} = 5.25\] The variance of this experiment is 5.25.