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When we talk about discrete random variables, we are referring to variables that can take on a countable number of distinct outcomes. In this exercise, the discrete random variable is the number of phone lines in use at any given time. The number of possible values for this variable ranges from 0 to 6, which corresponds to the total number of lines available. Each possible outcome is associated with a specific probability, indicating how likely it is to occur.

Discrete random variable outcomes can often be visualized with a probability distribution. This distribution displays all possible values the variable can assume along with their respective probabilities. In our example:

• 0 lines in use has a probability of 0.10
• 1 line in use has a probability of 0.15
• 2 lines in use has a probability of 0.20
• 3 lines in use has a probability of 0.25
• 4 lines in use has a probability of 0.20
• 5 lines in use has a probability of 0.06
• 6 lines in use has a probability of 0.04

Understanding discrete random variables helps to model and forecast outcomes in various fields, from telecommunication to finance.

The mean and standard deviation provide a summary of our probability distribution. Let's first consider the mean, which is essentially the average expected value of our random variable. It is calculated by multiplying each possible value by its probability and then adding all the results together. For our distribution, this results in a mean, or expected value, of 2.64. This means on average, about 2 to 3 phone lines are in use at any time.

The standard deviation, on the other hand, tells us how much variation or dispersion exists from the mean. It's crucial to understanding the spread of our data around this average. We first need the variance as an intermediate step, which is derived from the mean of the squares of each value minus the square of the mean.

• The variance, calculated as 80.0304, gives us an idea of the overall spread of the data.
• The standard deviation is the square root of the variance, giving us approximately 8.947.

This higher standard deviation indicates a large spread of values, suggesting that the number of lines in use can vary significantly from the mean.

Variance calculation is an important part of statistical analysis as it shows how much the values of a random variable differ from its mean. In our example, we began by computing the expected value of the square of each outcome, \(E(x^2)\), which involves squaring each possible value, multiplying by its probability, and summing them.

This result is then used to find the variance using the formula: \[Var(x) = E(x^2) - E(x)^2\]

• We calculated this to be 87 - 6.9696 = 80.0304.

The variance reflects the degree to which each probable value of the random variable deviates from the mean. A larger variance indicates that the data points are spread out over a wide range of values. In practical terms, this means there can be a significant fluctuation in the number of lines in use. By understanding variance, businesses can better anticipate and plan for peak usage times, potentially improving service efficiency and customer satisfaction. Variance is computed directly from probability formulas and provides a foundational measure for further statistical analysis, such as determining the probability of events concerning standard deviations.