Expectation and Variance of X_n

Expectation:
The expectation of a random variable is the average value it would take over a large number of trials. In this case, the random variable is X_n, which represents the sum of points obtained when n fair dice are rolled together.

To calculate the expectation of X_n, we need to consider the possible outcomes and their probabilities. When a fair die is rolled, there are six equally likely outcomes, ranging from 1 to 6. Therefore, the sum of points obtained when n fair dice are rolled together can range from n to 6n.

We can calculate the expectation of X_n using the formula:

E(X_n) = Σ(xi * P(xi))

where xi represents each possible outcome of X_n and P(xi) represents the probability of that outcome.

For example, when n = 1, the possible outcomes for X_n are {1, 2, 3, 4, 5, 6} with equal probabilities of 1/6 each. Therefore:

E(X_1) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5

Similarly, we can calculate the expectation for higher values of n, taking into account all the possible outcomes and their probabilities.

Variance:
The variance of a random variable measures how much the values of the variable vary around the expected value. It provides a measure of the spread or dispersion of the variable.

The variance of X_n can be calculated using the formula:

Var(X_n) = E((X_n - E(X_n))^2)

where E(X_n) represents the expectation of X_n.

To calculate the variance, we need to first calculate the squared deviation of each possible outcome from the expectation, and then multiply it by the probability of that outcome. Finally, we sum up all these values to get the variance.

For example, when n = 1, the possible outcomes for X_n are {1, 2, 3, 4, 5, 6} with equal probabilities of 1/6 each. The expectation of X_1 is 3.5, as calculated earlier.

Var(X_1) = ((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 * 1/6) ≈ 2.92

Similarly, we can calculate the variance for higher values of n, taking into account all the possible outcomes and their probabilities.

Conclusion:
In summary, the expectation of X_n represents the average value of the sum of points obtained when n fair dice are rolled together. The variance of X_n measures how much the values of X_n vary around its expected value, providing a measure of the spread or dispersion of the variable. By calculating the expectation and variance, we can gain insights into