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In probability theory, the mean of a discrete probability distribution represents the average outcome you would expect by repeating a random experiment many times. This is calculated through the expected value, denoted as \( E(X) \). It provides a measure of central tendency, such as how centralize or "typical" the measurement is from the given outcomes. For a discrete distribution, the mean is found using the formula: \[E(X) = \sum x_i P(x_i)\] This equation sums up each possible value \( x_i \) of a random variable multiplied by its corresponding probability \( P(x_i) \). Let's break it down using an example: for the distribution with outcomes 2, 8, and 10, and probabilities 0.5, 0.3, and 0.2 respectively, the expected value is calculated as follows:

• \( 2 \times 0.5 = 1.0 \)
• \( 8 \times 0.3 = 2.4 \)
• \( 10 \times 0.2 = 2.0 \)

Adding these values gives the expected mean value:\[E(X) = 1.0 + 2.4 + 2.0 = 5.4\] This means that if you could repeat the random experiment many times, you would expect the average outcome to be approximately 5.4.

The variance of a discrete probability distribution measures the spread or dispersion of possible outcomes around the mean (expected value). It tells you how much the values deviate from the mean, providing an understanding of their variability. Variance is computed using the formula: \[Var(X) = \sum (x_i - E(X))^2 P(x_i)\] Here, you subtract the mean \( E(X) \) from each possible outcome \( x_i \), square this difference, multiply by the corresponding probability, and then sum it all up. Let's understand this by applying it to our example:

• For \( x = 2 \) : \((2 - 5.4)^2 = 11.56\), \( 11.56 \times 0.5 = 5.78 \)
• For \( x = 8 \) : \((8 - 5.4)^2 = 6.76\), \( 6.76 \times 0.3 = 2.028 \)
• For \( x = 10 \) : \((10 - 5.4)^2 = 21.16\), \( 21.16 \times 0.2 = 4.232 \)

The sum of these calculations results in: \[Var(X) = 5.78 + 2.028 + 4.232 = 12.04\] Thus, the variance of 12.04 indicates the average squared deviation from the mean, reflecting the diversity or uncertainty in the distribution.

Probability theory is the branch of mathematics concerned with analyzing random phenomena and deciding how likely events are to occur. It provides the framework for understanding discrete probability distributions, which describe the probability of each possible outcome of a random process. Such distributions are based on two key elements: outcomes and probabilities.

• Outcomes: The possible results of a random process (like rolling a dice or drawing a card).
• Probabilities: Represent the likelihood of each outcome, summing up to 1 across all possibilities.

The core idea is that by systematically assigning probabilities to distinct outcomes and using statistical measures like mean and variance, probability theory helps us make informed predictions and decisions based on uncertainty. For example, knowing the mean tells us the expected result, while the variance highlights how much results may vary. Probability theory thereby forms the backbone of data science, risk assessment, and many fields that involve the analysis of random processes.