E(x) = σ [x * p(x)]

Definition

The expression e(x) = σ [x * p(x)] represents the expected value of a discrete random variable, which is a fundamental concept in probability theory. This formula shows that the expected value is calculated by summing the product of each possible outcome of the random variable (x) and its corresponding probability (p(x)). Understanding this concept is crucial for evaluating the average or typical outcome of random processes, as well as for calculating variance and moments of the random variable.

5 Must Know Facts For Your Next Test

1. The expected value e(x) provides a single number that summarizes the entire distribution of a discrete random variable.
2. To compute e(x), you multiply each outcome by its probability and then sum all these products.
3. If all outcomes are equally likely, e(x) simplifies to the average of those outcomes.
4. The expected value can be used in various applications, including finance, insurance, and risk assessment.
5. e(x) does not account for variability; it only provides an average outcome, which is why variance is often calculated alongside it.

Review Questions

• How do you compute the expected value of a discrete random variable using e(x) = σ [x * p(x)], and what does this computation reveal about the variable?
  • To compute the expected value using e(x) = σ [x * p(x)], you take each possible outcome of the discrete random variable and multiply it by its associated probability. Then, you sum all these products together. This computation reveals the average outcome you can expect when observing many instances of the random process represented by the variable. It gives a useful summary of what one might anticipate over time.
• Discuss how understanding e(x) is essential for calculating variance and moments for discrete random variables.
  • Understanding e(x) is crucial because variance and higher moments build upon the concept of expected value. Variance requires knowing e(x) to calculate how far each outcome deviates from this average, effectively measuring dispersion around it. Higher moments, like skewness and kurtosis, also rely on e(x) as they describe other aspects of the distribution's shape. Thus, e(x) serves as a foundational tool in statistical analysis.
• Evaluate how changes in probabilities affect the expected value e(x) and what implications this may have in real-world scenarios.
  • Changes in probabilities directly influence e(x), as each outcome's contribution to the expected value is weighted by its probability. If an outcome with a high payoff increases in probability, the expected value will rise, which might indicate a better investment or decision-making scenario. Conversely, if low-probability high-loss events gain more weight, it could signal increased risk. This sensitivity highlights the importance of assessing probabilities accurately in fields like finance or decision theory.

Related terms

Random Variable: A variable whose possible values are numerical outcomes of a random phenomenon.

Probability Mass Function: A function that gives the probability that a discrete random variable is equal to a specific value.

Variance: A measure of the spread or dispersion of a set of values, calculated as the average of the squared differences from the mean.