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The expected value of a random variable is a fundamental concept in probability and statistics. It gives us a measure of the central tendency, or the "average" value, that we can expect from a probability distribution. Think of it as a way to summarize the distribution of possible outcomes for a random variable. To find the expected value for a discrete random variable, you multiply each possible outcome by its probability and then sum these products.In our exercise, the random variable, denoted as \(X\), can take on the values 1, 2, 3, and 4 with respective probabilities of 0.4, 0.3, 0.2, and 0.1. The formula to compute the expected value is:

• \(E(X) = \sum x_i P(X = x_i)\)

Here, \(E(X)\) is calculated as:

• \(E(X) = (1 \cdot 0.4) + (2 \cdot 0.3) + (3 \cdot 0.2) + (4 \cdot 0.1) = 2\)

This means that, on average, we would expect the value of \(X\) to be 2 over many trials.

Variance provides a measure of how much the values in a probability distribution spread out from the expected value. It tells us how much variance, or "spread," there is from the mean. A higher variance indicates that the numbers are more spread out.To compute variance, we first must find \(E(X^2)\), which represents the expected value of the square of the random variable. This calculation involves squaring each outcome, multiplying by its probability, and summing the results:

• \(E(X^2) = (1^2 \cdot 0.4) + (2^2 \cdot 0.3) + (3^2 \cdot 0.2) + (4^2 \cdot 0.1) = 5.6\)

Then, the variance \(Var(X)\) is calculated using:

• \(Var(X) = E(X^2) - [E(X)]^2 = 5.6 - 2^2 = 1.6\)

This means that \(X\)'s values vary by approximately 1.6 from the square of the mean.

The standard deviation is a widely used measure of variability that quantifies the amount of variation or dispersion in a set of values. It is the square root of the variance and provides a measure that is in the same units as the data, making it more interpretable.To find the standard deviation from the variance, you simply take the square root:

• \( \sigma(X) = \sqrt{Var(X)} \)
• \( \sigma(X) = \sqrt{1.6} \approx 1.26\)

Thus, the standard deviation gives us a sense of how much the values of \(X\) typically deviate from the mean value of 2. In practical terms, a standard deviation of approximately 1.26 means that the values of \(X\) tend to vary by about 1.26 units from the mean.