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The mean, or expected value, of a random variable provides a measure of the center of its distribution. It's essentially the weighted average of all possible values that the random variable can take, with the weights being the probabilities of each value. To find the mean of a random variable \(X\), you use the formula:\[E[X] = \sum x \cdot P(X = x)\]This formula summates the products of each value \(x\) and its probability \(P(X = x)\). By multiplying each value by its probability and summing these products, you're essentially averaging out what value you "expect" the random variable \(X\) to take.

• This involves all possible outcomes of \(X\).
• It's crucial in probability as it provides essential insight into the distribution's characteristics.

In our problem, after following this process, we found that the mean \(E[X]\) is -191.5. This indicates that, on average, the random variable \(X\) tends to lean towards this value, considering the entire distribution.

Variance measures the spread of a set of values. It tells us how much the values deviate from the mean, on average. To calculate variance for the random variable \(X\), we start by finding \(E[X^2]\), which is the expected value of the square of \(X\). The calculated formula is:\[E[X^2] = \sum x^2 \cdot P(X = x)\]Then, using \(E[X^2]\) and the mean \(E[X]\), the variance \(\text{Var}(X)\) is discovered through:\[\text{Var}(X) = E[X^2] - (E[X])^2\]

• The term \(E[X^2]\) accounts for the distribution of values and their potential values—each squared and then averaged by their probabilities.
• The subtraction of \((E[X])^2\) adjusts this spread to focus on variations around the mean.

In our example, the variance \(\text{Var}(X)\) was finally ascertained to be 74.05, indicating the spread or dispersion of \(X\) around its mean.

The standard deviation is an extension of the concept of variance, offering a straightforward measure of dispersion in the same units as the original data. It essentially represents the average distance of data points from the mean. To find the standard deviation \(\sigma(X)\), you take the square root of the variance:\[\sigma(X) = \sqrt{\text{Var}(X)}\]

• This transformation provides a more interpretable measure of variability than variance alone, as it returns the measure to the same scale as the original data.
• It's widely used as it is easier to understand in terms of real-world data analysis.

For our probability distribution, the standard deviation \(\sigma(X)\) is approximately 8.6. This implies that, on average, the values of \(X\) deviate from the mean by about 8.6 units, offering insight into the typical spread of the data set.