91Ó°ÊÓ

A random variable is a fundamental concept in statistics and probability theory. It's essentially a variable that can take on different numerical values, each associated with a probability based on a random event. To further simplify, think of a random variable as a way to map outcomes of a random process to numbers. For Seaside Villas, Inc., we defined the random variable \( X \) to represent the number of vacant villas in a month. Various outcomes from this random variable occur with certain probabilities. These outcomes were:\[\begin{align*}&X = 0 \text{ with } P(X = 0) = 0.1 \&X = 1 \text{ with } P(X = 1) = 0.2 \&X = 2 \text{ with } P(X = 2) = 0.3 \&X = 3 \text{ with } P(X = 3) = 0.4 \\end{align*}\] This setup helps us understand the distribution and likelihood of different scenarios, such as no vacant villas, one vacant, and so on.

The expected value, often referred to as the mean, is a key concept in statistics representing the average of all possible outcomes. It provides a central measure that can predict the typical result if the process was repeated multiple times.To calculate the expected value \( E(X) \) for the number of vacant villas, we use the formula:\[ E(X) = \sum x_i \cdot P(x_i) \]By substituting our values, we obtain:

• For \( x = 0 \), the calculation is \( 0 \times 0.1 = 0 \)
• For \( x = 1 \), the calculation is \( 1 \times 0.2 = 0.2 \)
• For \( x = 2 \), the calculation is \( 2 \times 0.3 = 0.6 \)
• For \( x = 3 \), the calculation is \( 3 \times 0.4 = 1.2 \)

Summing these gives us \( E(X) = 0 + 0.2 + 0.6 + 1.2 = 2.0 \). This means, on average, there are 2 vacant villas each month.

Variance is an important measure in statistics that shows the degree of spread or dispersion of a set of values. It reflects how much the outcomes deviate from the expected value (mean), providing insights into the variability within the random variable.The formula for variance is:\[ \text{Var}(X) = \sum \left((x_i - E(X))^2 \cdot P(x_i)\right) \]Let's determine \( \text{Var}(X) \) for the vacant villas scenario:

• For \( x = 0 \), \((0 - 2)^2 = 4\) then \(4 \cdot 0.1 = 0.4\)
• For \( x = 1 \), \((1 - 2)^2 = 1\) then \(1 \cdot 0.2 = 0.2\)
• For \( x = 2 \), \((2 - 2)^2 = 0\) then \(0 \cdot 0.3 = 0\)
• For \( x = 3 \), \((3 - 2)^2 = 1\) then \(1 \cdot 0.4 = 0.4\)

Adding these terms gives \( \text{Var}(X) = 0.4 + 0.2 + 0 + 0.4 = 1.0 \). This variance value tells us that the fluctuation of vacant villas around the mean is moderate.

The standard deviation is a statistical measure that provides insight into the amount of variation or dispersion in a set of values. It is the square root of the variance, giving us a measure that remains in the same units as the data, making it easily interpretable.For the case of the vacant villas, the variance \( \text{Var}(X) = 1.0 \). To find the standard deviation \( \sigma \), we compute:\[ \sigma = \sqrt{\text{Var}(X)} = \sqrt{1.0} = 1.0 \]This tells us that the number of vacant villas typically deviates by about 1 villa from the mean of 2 vacant villas each month. The standard deviation helps in understanding the consistency or reliability of the observed vacancies from month to month.