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To find the mean, or expected value, of a probability distribution, you need to multiply each possible outcome (in this case, the number of systems sold) by its corresponding probability, and then sum all these products. The formula for the mean \(\mu\) is given by: \[ \mu = \sum x \cdot p(x) \]In simple terms, this gives you the average or expected number of systems sold based on the probabilities from the distribution. In this exercise, you calculate the mean through the given probabilities, helping to predict sales patterns.

This concept is crucial because it acts as a balancing point, offering insight into what you can typically expect over a period of time. If you had to make a business decision, knowing the mean helps in planning and setting realistic targets.

Variance and standard deviation are important measures that tell us about the spread of a probability distribution. While the mean tells us about the central tendency, variance quantifies how much the numbers are spread out.
Variance is calculated by taking each outcome, subtracting the mean, squaring the result, and then multiplying by its probability:\[ \sigma^2 = \sum (x-\mu)^2 \cdot p(x) \]The standard deviation is simply the square root of the variance:\[\sigma = \sqrt{\sigma^2}\]Together, they give a clearer picture of variability. A high variance and standard deviation mean that the number of systems sold each month varies widely from the mean.

Conversely, low values hint that sales numbers are consistently close to the mean. Understanding these helps businesses assess risk and predict fluctuations.

Standard deviation is a key concept in understanding data spread. It tells us how much variation exists from the mean. The closer the values are to the mean, the smaller the standard deviation and vice versa.
It's not just about numbers; interpreting standard deviation provides insights into stability and predictability.

• If the number of systems sold is mostly within 1 standard deviation of the mean, it means sales are stable.
• If results often fall beyond 2 standard deviations, it indicates larger fluctuations or potential outliers.

For instance, when predicting sales, knowing the standard deviation can help a business plan for variability and manage expectations. This interpretation aids in setting realistic goals and in risk management.