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Calculating the mean of a probability distribution is a straightforward process that provides the average outcome you might expect in the long run. In the context of our exercise, the mean gives us an idea of how many people on average are using the Mountain Climber machine. To find the mean, you multiply each possible outcome by its corresponding probability and sum these products. This is represented by the formula \( \mu = \sum X_i P(X_i) \), where \( X_i \) represents the number of occurrences, and \( P(X_i) \) is the probability of \( X_i \).
For our exercise, the mean is calculated as:
- \( (0 \times 0.1) \) for 0 users.- \( (1 \times 0.2) \) for 1 user.- \( (2 \times 0.4) \) for 2 users.- \( (3 \times 0.2) \) for 3 users.- \( (4 \times 0.1) \) for 4 users.Adding these values: \( 0 + 0.2 + 0.8 + 0.6 + 0.4 = 2.0 \). So, the mean number of users is 2. This means that, on average, 2 people use the machine over a 3-hour period.

Understanding variance is essential as it measures the spread of a probability distribution around the mean. It gives insights into how much the values differ from the average, indicating the consistency of the outcomes. The variance \( \sigma^2 \) is calculated by taking the sum of the squared differences between each possible outcome and the mean, each weighted by its probability. This is written mathematically as \( \sigma^2 = \sum (X_i - \mu)^2 P(X_i) \).
In our exercise:

• For 0 users: \((0 - 2)^2 \times 0.1 = 0.4\)
• For 1 user: \((1 - 2)^2 \times 0.2 = 0.2\)
• For 2 users: \((2 - 2)^2 \times 0.4 = 0\)
• For 3 users: \((3 - 2)^2 \times 0.2 = 0.2\)
• For 4 users: \((4 - 2)^2 \times 0.1 = 0.4\)

Summing these gives a variance of \( 1.2 \). This figure reflects that the number of people using the machine varies, but not excessively when viewed in the context of the entire distribution.

Standard deviation is a vital statistical tool that converts the variance into a more interpretable metric. It represents an average distance from the mean for each value in a data set. By taking the square root of the variance, the standard deviation provides a measure of spread that is in the same unit as the original data. This makes it easier to understand and communicate variability.
For the exercise involving the Mountain Climber machine, the variance was found to be \( 1.2 \). The standard deviation \( \sigma \) is therefore calculated as \( \sigma = \sqrt{1.2} \). Using a calculator, this gives approximately \( 1.095 \).
This value indicates that the typical deviation from the average number of people using the machine (which is 2) is about 1.095. Such a measure helps in appreciating how consistent the usage pattern over the 3-hour observation period is. So, while on average 2 people use the machine, the number might vary by about 1 person under typical circumstances.