Understanding the Problem: Variance of a Die Roll

The question asks for the variance of the number shown on the uppermost face when a fair six-sided die is rolled. This involves understanding probability distributions and how to calculate variance for a discrete random variable.

Defining the Random Variable and Distribution

Let \(X\) be the random variable representing the number on the uppermost face when a fair six-sided die is rolled. The possible outcomes for \(X\) are the integers from 1 to 6.

Since the die is fair, each outcome has an equal probability. This is a discrete uniform distribution. The probability mass function is:

• \(P(X=x) = \dfrac{1}{6}\) for \(x \in \{1, 2, 3, 4, 5, 6\}\)
• \(P(X=x) = 0\) otherwise

We can represent the distribution in a table:

Calculating Expected Value (\(E[X]\))

The expected value, or mean, of a discrete random variable \(X\) is given by the formula:

$$E[X] = \sum x \cdot P(X=x)$$

For the fair six-sided die:

$$E[X] = \left(1 \cdot \dfrac{1}{6}\right) + \left(2 \cdot \dfrac{1}{6}\right) + \left(3 \cdot \dfrac{1}{6}\right) + \left(4 \cdot \dfrac{1}{6}\right) + \left(5 \cdot \dfrac{1}{6}\right) + \left(6 \cdot \dfrac{1}{6}\right)$$

$$E[X] = \dfrac{1}{6} (1 + 2 + 3 + 4 + 5 + 6) = \dfrac{1}{6} (21) = \dfrac{21}{6} = \dfrac{7}{2} = 3.5$$

The expected value of rolling a fair six-sided die is 3.5.

Calculating Variance (\(Var(X)\))

The variance of a random variable \(X\) measures how spread out the possible outcomes are from the expected value. The formula for variance is:

$$Var(X) = E[X^2] - (E[X])^2$$

First, we need to calculate \(E[X^2]\). This is the expected value of the square of the random variable:

$$E[X^2] = \sum x^2 \cdot P(X=x)$$

For the fair six-sided die:

$$E[X^2] = \left(1^2 \cdot \dfrac{1}{6}\right) + \left(2^2 \cdot \dfrac{1}{6}\right) + \left(3^2 \cdot \dfrac{1}{6}\right) + \left(4^2 \cdot \dfrac{1}{6}\right) + \left(5^2 \cdot \dfrac{1}{6}\right) + \left(6^2 \cdot \dfrac{1}{6}\right)$$

$$E[X^2] = \dfrac{1}{6} (1 + 4 + 9 + 16 + 25 + 36) = \dfrac{1}{6} (91) = \dfrac{91}{6}$$

Now, we can calculate the variance using the formula \(Var(X) = E[X^2] - (E[X])^2\):

$$Var(X) = \dfrac{91}{6} - \left(\dfrac{7}{2}\right)^2$$

$$Var(X) = \dfrac{91}{6} - \dfrac{49}{4}$$

To subtract these fractions, we find a common denominator, which is 12:

$$Var(X) = \dfrac{91 \cdot 2}{6 \cdot 2} - \dfrac{49 \cdot 3}{4 \cdot 3}$$

$$Var(X) = \dfrac{182}{12} - \dfrac{147}{12}$$

$$Var(X) = \dfrac{182 - 147}{12}$$

$$Var(X) = \dfrac{35}{12}$$

The variance of the number on the uppermost face of a fair six-sided die is \(\dfrac{35}{12}\).

Verification of the Variance Calculation

Let's review the steps taken to calculate the variance:

1. Identified the possible outcomes for rolling a fair six-sided die.
2. Determined the probability for each outcome (1/6 for each).
3. Calculated the expected value \(E[X]\).
4. Calculated the expected value of \(X\) squared, \(E[X^2]\).
5. Used the variance formula \(Var(X) = E[X^2] - (E[X])^2\).
6. Performed the final subtraction to get \(\dfrac{35}{12}\).

The calculation is consistent and follows the standard method for finding the variance of a discrete probability distribution.

Conclusion on Variance of Die Roll

Based on the calculations, the variance of the number on the uppermost face when a fair six-sided die is rolled is \(\dfrac{35}{12}\).

Revision Table: Key Concepts

Additional Information: Variance and Standard Deviation

Variance is an important measure of dispersion in statistics. A higher variance indicates that the data points are more spread out from the mean, while a lower variance indicates they are clustered closer to the mean.

The standard deviation is the square root of the variance. It is often used because it is in the same units as the random variable itself, making it easier to interpret than variance.

For this problem, the standard deviation would be \(\sqrt{\dfrac{35}{12}}\).

Understanding expected value and variance is fundamental in probability and statistics for analyzing the properties of random variables and probability distributions.