Final answer:
The mean, variance, and standard deviation of the face value of a fair die can be calculated by considering the probability of each face value. The mean is 3.5, the variance is 35/12, and the standard deviation is approximately 1.71.
Step-by-step explanation:
To find the mean, variance, and standard deviation of the face value of a fair die, we first need to determine the probability of each face value occurring. Since a fair die has six sides, each side has an equal probability of 1/6.
To find the mean, we multiply each face value by its probability and add them together. The mean = (1/6) * 1 + (1/6) * 2 + (1/6) * 3 + (1/6) * 4 + (1/6) * 5 + (1/6) * 6 = 3.5.
To find the variance, we need to calculate the squared difference between each face value and the mean, multiplied by its probability, and add them together. The variance = (1/6) * (1-3.5)^2 + (1/6) * (2-3.5)^2 + (1/6) * (3-3.5)^2 + (1/6) * (4-3.5)^2 + (1/6) * (5-3.5)^2 + (1/6) * (6-3.5)^2 = 35/12.
To find the standard deviation, we simply take the square root of the variance. The standard deviation = sqrt(35/12) ≈ 1.71.