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The mean of a probability distribution, also known as the expected value, is a central concept in statistics. It gives us a single number that represents the average outcome of a random process. In the context of our exercise, the mean tells us how many camcorders the electronics store can expect to sell on any given day.

To compute the mean, we multiply each possible number of camcorders sold (discrete outcomes) by its probability and then sum them up. This provides a weighted average where each outcome's weight is its probability. For example, in our problem, the mean is calculated using the formula:

\[ \text{Mean} = \sum_{i=0}^{6} X_i \times P(X_i) \]

Where \(X_i\) represents the number of camcorders sold, and \(P(X_i)\) is the probability of selling \(X_i\) camcorders. This approach ensures that outcomes that are more likely have a bigger impact on the mean.

The standard deviation gives us a measure of how much each possible outcome deviates from the mean. It is a critical aspect of understanding how data is distributed around the mean.

To find the standard deviation, we first need to calculate the variance. Variance tells us the average of the squared deviations from the mean, providing insight into the data's spread. The formula to calculate variance is:

\[ \sigma^2 = \sum_{i=0}^{6} (X_i - \text{mean})^2 \times P(X_i) \]

Once we have the variance, the standard deviation is simply the square root of this value:

\[ \sigma = \sqrt{\sigma^2} \]

By knowing the standard deviation, we can understand the consistency of camcorder sales. A smaller standard deviation indicates that sales figures are closely packed around the mean, while a larger one suggests higher variability.

A discrete random variable is a type of variable that can take on a countable number of distinct values. In our exercise, the number of camcorders sold each day is a discrete random variable because it can only take integer values such as 0, 1, 2, etc. It cannot take fractional values like 2.5.

Discrete random variables are often accompanied by a probability distribution, which specifies the probabilities of each potential value. The distribution must satisfy two conditions:

• The probabilities of all possible outcomes must sum up to 1.
• Each probability must be between 0 and 1, inclusive.

When working with discrete random variables, we can calculate important statistical measures like the mean, variance, and standard deviation to better understand the nature of the data.

Variance is a fundamental concept in statistics that describes how far the values of a random variable spread out from the mean. It gives us an idea of the data's variability.

In our probability distribution of camcorders sold, variance is calculated using the formula:

\[ \sigma^2 = \sum_{i=0}^{6} (X_i - \text{mean})^2 \times P(X_i) \]

Here each outcome \((X_i)\) is subtracted from the mean, the difference is squared, and then multiplied by the probability of that outcome. This results in all deviations being converted into positive values, which are weight-adjusted by their probability.

The final sum gives an average of these squared deviations, showing us how much variation exists around the mean. A small variance indicates that the values are closer to the mean, suggesting predictable and consistent sales. In contrast, a large variance points towards a broader range of sales figures.