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In the realm of probability, understanding a discrete random variable is foundational. A discrete random variable is one that can take on a countable number of distinct values. This contrasts with continuous random variables, which can take on an infinite number of values within a range.
For instance, a die's roll is a perfect example of a discrete random variable, as it can only result in one of the six natural numbers (1 through 6). Our current exercise uses values of 0, 1, 2, and 3.
The probabilities associated with each outcome give us more insights. These probabilities must satisfy two key conditions:

• Each probability must be between 0 and 1.
• The sum of all probabilities must add up to 1.

Understanding these concepts ensures that any calculations regarding random variables, such as mean and variance, are accurate and relevant.

The mean of a discrete random variable, also known as the expected value, provides insight into the 'average' outcome if an experiment were repeated many times.
The formula for calculating the mean, or expected value, is represented by:

• \[E(X) = \sum x_i P(x_i)\]

To break it down, each value of the random variable (\(x_i\)) is multiplied by its probability (\(P(x_i)\)).
The results are then summed to arrive at the expected value. For our exercise:

• 0 times 0.2 equals 0.
• 1 times 0.4 equals 0.4.
• 2 times 0.3 equals 0.6.
• 3 times 0.1 equals 0.3.

Adding these gives a mean of 1.3. Ultimately, the mean gives us a central tendency of the distribution, helping to predict general outcomes.

Calculating variance allows us to understand how much the values of a discrete random variable will spread around the mean.
Variance is mathematically represented by:

• \[Var(X) = \sum (x_i - E(X))^2 P(x_i)\]

This involves three main steps:

• Subtracting the mean from each discrete value to find the deviation.
• Squaring each deviation to eliminate negative values and emphasize larger distances.
• Multiplying each squared deviation by the corresponding probability, then summing all these values.

Example computations from our exercise include:

• \((0 - 1.3)^2 \times 0.2 = 0.338\)
• \((1 - 1.3)^2 \times 0.4 = 0.036\)
• \((2 - 1.3)^2 \times 0.3 = 0.147\)
• \((3 - 1.3)^2 \times 0.1 = 0.289\)

The sum of these computations gives us a variance of 0.81. Variance tells us about the consistency of the data—it quantifies the variability.

The expected value is a critical concept in statistics and probability, acting as the cornerstone for predicting likely outcomes.
This value, represented by the mean of the distribution, helps forecast what might happen in a random experiment, given sufficiently many repetitions.
Mathematically, the expected value of a discrete random variable is consistent across calculations:

• It averages out the possible outcomes by their respective occurrences (probabilities).
• It provides a guide rather than an assured outcome, as results for single events can significantly differ.

In our exercise, the expected value is computed to be 1.3, signifying that over many trials, the average result will tend toward this number. It serves as a powerful tool in decision-making and risk assessment across a variety of fields, from finance to engineering.