91Ó°ÊÓ

Understanding the mean and standard deviation of a probability distribution is essential in statistics. The mean, also known as the expected value, gives us an idea of the central tendency of the data. In the case of our freezer example, the mean storage space can be calculated using the formula:

• \(E(x) = \sum x \cdot p(x) = (13.5 \cdot 0.2) + (15.9 \cdot 0.5) + (19.1 \cdot 0.3)\)

This tells us the average amount of storage space purchased. To calculate standard deviation, we first find the variance, which measures how much the values differ from the mean. The variance is given by:

• \(var(x) = \sum ([x - E(x)]^2 \cdot p(x))\)

After finding the variance, the standard deviation is simply the square root of this variance. It indicates the spread of values around the mean. A smaller standard deviation means the data points are close to the mean, while a larger one indicates more spread. In our problem, after calculating variance, you take its square root to get the standard deviation.

The concept of expected value is central to understanding probability distributions. It is the long-run average of repetitions of the experiment it represents. For any given random variable like storage space or pricing, the expected value can be thought of as a weighted average of all possible values the variable can take on. This is where each outcome is weighted by its probability of occurrence.

If you consider the price of a freezer based on storage size, the expected price paid by a customer is a key insight. We can say:

• For a price defined by \(Price = 25x - 8.5\), the expected price is calculated as: \(E(Price) = 25 \times E(x) - 8.5\).

This equation uses the linearity property of expectation, meaning the expected value operation distributes over addition and scalar multiplication. Thus, you use the expected storage size from earlier to find the expected price.

Variance quantifies how much the values of a random variable differ from the expected value. It provides a measure of the distribution's "spread." In simpler terms, variance answers: "How much do the storage sizes fluctuate around the mean?" The greater the variance, the more spread out the values are.

When you derive the variance for price, it's important to remember that if a random variable is multiplied by a constant, the variance will be affected by the square of that constant. For the price, the variance is:

• \(var(Price) = (25^2) \cdot var(x)\)

The formula shows that scaling the original variable significantly amplifies its variance. This is crucial for determining how uncertain or variable the pricing can be, and understanding it helps in appraising the financial aspect of freezer sales.