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Expected value is a fundamental concept in statistics that represents the average outcome of a random variable over a large number of trials. To calculate the expected value (mean) in this problem, you need to multiply the average number of patient records stolen per breach by the total number of breaches.

Using the formula:
\[ E(\text{total}) = n \times \bar{x} \]
where:

• \( n \) = number of breaches
• \( \bar{x} \) = average number of records stolen per breach

Given the values 45 breaches and 361 records per breach, substituting them in:
\[ E(\text{total}) = 45 \times 361 = 16,245 \]
Thus, the expected value of the number of patient records stolen during 45 breaches is 16,245.

Standard deviation measures the amount of variation or dispersion in a set of values. In this case, it helps understand the variability in the number of patient records stolen per breach. When dealing with multiple independent events, the total standard deviation can be calculated using the square root of the number of breaches multiplied by the standard deviation per breach.

The formula is:
\[ \text{SD(\text{total})} = \sqrt{n} \times \sigma \]
where:

• \( n \) = number of breaches
• \( \sigma \) = standard deviation per breach

Given the values 45 breaches and a standard deviation of 74, substituting them in:
\[ \text{SD(\text{total})} = \sqrt{45} \times 74 \approx 496.41 \]
Thus, the total standard deviation for the number of records stolen is approximately 496.41.

In probability and statistics, events are said to be independent if the occurrence of one event does not affect the probability of occurrence of another event. Understanding the independence of events is crucial when applying statistical formulas.

In this exercise, each breach is assumed to be an independent event. This means the outcome of one breach (the number of records stolen) does not influence the outcome of another breach.
Because of this independence, standard statistical formulas for sums of variables can be applied directly. We use the given average and standard deviation values per breach and then scale them up for 45 independent breaches using relevant formulas.
This simplifies calculations and ensures accurate statistical results.

Statistical formulas are vital tools in analyzing data. They provide methods to calculate key metrics like mean and standard deviation. In this problem, we use specific formulas to find the expected value and standard deviation for the number of records stolen.

The important formulas are:

• Expected value (mean):
  \[ E(\text{total}) = n \times \bar{x} \]
• Standard deviation:
  \[ \text{SD(\text{total})} = \sqrt{ n} \times \sigma \]

These formulas are derived from basic principles of statistics and apply to scenarios involving sums of random variables, especially in cases of independent events. They allow us to efficiently compute the desired outcomes based on the input data given.