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The variance of X, the face value of a fair 10-sided die is:
(select the answer closest to yours) 385, 38.5, 8.25, 2.917, 5.5

Answer :

To find the variance of [tex]X[/tex], the face value of a fair 10-sided die, we can follow these steps:


  1. Identify the possible outcomes: A 10-sided die has face values from 1 to 10, so the outcomes are [tex]1, 2, 3, ..., 10[/tex].


  2. Calculate the mean (expected value) [tex]\mu[/tex]: Since the die is fair, each face has an equal probability of [tex]\frac{1}{10}[/tex]. The mean can be calculated using the formula for the expected value:

    [tex]\mu = \sum_{i=1}^{10} x_i \cdot P(x_i) = \frac{1}{10}(1 + 2 + 3 + \cdots + 10)[/tex]

    The sum of the numbers from 1 to 10 can be calculated using the formula for the sum of an arithmetic series:

    [tex]\sum_{i=1}^{10} i = \frac{10}{2} \times (1 + 10) = 5 \times 11 = 55[/tex]

    Therefore, the mean is:

    [tex]\mu = \frac{55}{10} = 5.5[/tex]


  3. Calculate the variance [tex]\sigma^2[/tex]: Variance is the expected value of the squared deviations from the mean. This can be calculated as:

    [tex]\sigma^2 = \sum_{i=1}^{10} (x_i - \mu)^2 \cdot P(x_i)[/tex]

    [tex]\sigma^2 = \frac{1}{10}[(1 - 5.5)^2 + (2 - 5.5)^2 + (3 - 5.5)^2 + \cdots + (10 - 5.5)^2][/tex]

    Calculating each squared deviation:

    [tex](1 - 5.5)^2 = 20.25, \quad (2 - 5.5)^2 = 12.25, \quad (3 - 5.5)^2 = 6.25, \quad (4 - 5.5)^2 = 2.25, \quad (5 - 5.5)^2 = 0.25[/tex]
    [tex](6 - 5.5)^2 = 0.25, \quad (7 - 5.5)^2 = 2.25, \quad (8 - 5.5)^2 = 6.25, \quad (9 - 5.5)^2 = 12.25, \quad (10 - 5.5)^2 = 20.25[/tex]

    Sum of these squared deviations:

    [tex]20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 82.5[/tex]

    Finally, divide by the number of outcomes to find the variance:

    [tex]\sigma^2 = \frac{82.5}{10} = 8.25[/tex]



Therefore, the variance of [tex]X[/tex], the face value of a fair 10-sided die, is [tex]8.25[/tex]. The closest answer from the choices provided is 8.25.

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