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When we say that two random variables, such as the amount of Type I chemical \(Y_1\) and the amount of Type II chemical \(Y_2\), are independent, we mean the outcome of one variable doesn't affect the outcome of the other. This is an important property in probability and statistics because it simplifies calculations. For example, when calculating the variance of the total cost, we can use the formula for the sum of variances only because \(Y_1\) and \(Y_2\) are independent.

Here are the key points about independent random variables:

• The joint probability distribution of independent variables is the product of their individual distributions.
• Covariance between independent random variables is zero.
• Independence is a stronger condition than uncorrelatedness; uncorrelated variables might not be independent.

Understanding this concept aids in realizing why the variance formula in our problem only involves the sum of individual variances without any covariance term.

The mean and variance are central concepts in statistics, providing measures of the central tendency and spread of a distribution, respectively. In our example, the mean (expected value) and variance for each type of chemical purchase are given.

The mean, or expected value, is a weighted average, representing the long-term average if the event were repeated many times. For Type I and Type II chemicals, the means are 40 gallons and 65 gallons, respectively. These values tell us what we would expect to happen on average each week.

The variance provides a measure of how much the data is spread out. A higher variance indicates a wider spread - more unpredictability. In our exercise, \( V(Y_1) = 4 \) and \( V(Y_2) = 8 \), indicating the purchases of Type II chemical vary more than those of Type I chemicals.

When calculating the total cost's mean and variance, these properties are essential as they allow us to obtain:

• The expected total cost is simply the sum of the expected individual costs.
• The variance of the total cost is the sum of the variances, provided the variables are independent.

Thus, applying these concepts, the expected total cost was calculated as \( \$445 \) with a variance of \( 236 \).

Cost analysis in statistics involves determining the expected costs and uncertainties associated with them. This is usually applied in business and economic fields to help in financial planning and decision-making. Our exercise is a classic example where cost analysis is used to anticipate the expenses for purchasing chemicals weekly.

In order to perform a statistical cost analysis, follow these steps:

• Identify the costs involved and represent them with variables.
• Calculate the expected cost by considering the prices and the expected amount purchased (mean of random variables).
• Assess the uncertainty or risk by computing the variance, giving insight into the cost volatility and unpredictability.

In practical terms, knowing the expected spending helps in budgeting and resource allocation, while understanding the variance can aid in risk management. A higher variance in costs could suggest a need for a contingency plan. Thus, cost analysis not only tells us what to expect but also prepares us for uncertainties.